Chapter 2 Nuclear Reactor Calculations

Chapter 2

Nuclear Reactor Calculations

Keisuke Okumura, Yoshiaki Oka, and Yuki Ishiwatari

Abstract The most fundamental evaluation quantity of the nuclear design

calculation is the effective multiplication factor (keff) and neutron flux distribution.

The excess reactivity, control rod worth, reactivity coefficient, power distribution,

etc. are undoubtedly inseparable from the nuclear design calculation. Some quan-

tities among them can be derived by secondary calculations from the effective

multiplication factor or neutron flux distribution. Section 2.1 treats the theory and

mechanism to calculate the effective multiplication factor and neutron flux distri-

bution in calculation programs (called codes). It is written by Keisuke Okumura.

The nuclear reactor calculation is classified broadly into the reactor core calcu-

lation and the nuclear plant characteristics calculation. The former is done to clarify

nuclear, thermal, or their composite properties. The latter is done to clarify dynamic

and control properties, startup and stability, and safety by modeling pipes and

valves of the coolant system, coolant pump, their control system, steam turbine

and condenser, etc. connected with the reactor pressure vessel as well as the reactor

core. The reactor core, plant dynamics, safety analysis and fuel rod analysis are

described in Sect. 2.2. It is written by Yoshiaki Oka and Yuki Ishiwatari.

2.1 Nuclear Design Calculations

2.1.1 Fundamental Neutron Transport Equation

The collective behavior of neutrons in a reactor core is described by the neutron

transport equation presented in Eq. (2.1) which is also referred to as the Boltzmann

equation.

Y. Oka (ed.), Nuclear Reactor Design, An Advanced Course in Nuclear Engineering 2,

DOI 10.1007/978-4-431-54898-0_2, © Authors 2014

49

ð2:1Þ

Here, S is the neutron source, Σt is the macroscopic total cross section, and ϕ is

the angular neutron flux being calculated. This equation represents the balance

between gain and loss in the unit volume of neutrons that are characterized by a

specific kinetic energy E (velocity υ) and are traveling in a specific direction ~Ω at a

time t and a position~r. That is, the time change of the target neutrons (the first term

in the LHS) is given by the balanced relation among: the gain of neutrons appearing

from the neutron source S (the first in the RHS), the net loss of neutrons traveling

(the second term in the RHS), and the loss of neutrons due to nuclear collisions (the

third term in the RHS). It should be noted that the changes in angle and energy of

neutrons are also included in the gain and loss.

The target neutrons are gained from three mechanisms: scattering, fission, and

external neutron sources. Each gain is represented in

ð2:2Þwhere Σs and Σf are the macroscopic scattering and fission cross sections, respec-

tively. v is the average number of neutrons released per fission and the product vΣf is

treated as a production cross section. The cross sections are described in the next

section.

The first term in the RHS of Eq. (2.2) is called the scattering source and it totals

the number of the target neutrons scattering into E and~Ω from another energy E0 and

direction ~Ω0by integrating the number for E0 and ~Ω

0. The second term is the fission

source and it indicates that the neutrons produced by fission over the whole range of

energies are distributed with the isotropic probability in direction (1/4π) and the

probability χ(E) in energy. χ(E) is called the fission spectrum and it is dependent on

the nuclide undergoing fission and the energy of incident neutrons. For instance, the

fission spectrum in an enriched uranium-fueled LWR is well described by the

function of Eq. (2.3).

ð2:3Þ

Since χ(E) is a probability distribution function, it is normalized so that

ð2:4Þ

50 K. Okumura et al.

The third term in the RHS of Eq. (2.2), Sex, expresses the external neutron sourcefor reactor startup which may be such species as 252Cf or Am-Be. Therefore, it is

not used in the nuclear design calculation of a reactor in operation.

More details of the neutron transport equation are not handled here. If the cross

section data (Σt, Σs, and vΣf) in Eqs. (2.1) and (2.2) are provided, the angular

neutron flux ϕ ~r; ~Ω;E; t� �

, which depends on location, traveling direction, energy,

and time, can be calculated by properly solving the equation.

The information on traveling direction of neutrons is finally unimportant in the

nuclear design calculation. The scalar neutron flux integrated over the angle is

rather meaningful.

ð2:5Þ

2.1.2 Neutron Spectrum

The function characterizing the energy dependence of neutron flux ϕ ~r;E; tð Þis called the neutron spectrum and it varies with fuel enrichment, moderator

density, void fraction, burnup, and so on. Figure 2.1 shows a neutron spectrum of

a thermal reactor. The neutron spectrum of thermal reactors is divided into three

distinct energy regions. In the high-energy region above 105 eV, since the prompt

neutrons released by fission are dominant, the neutron spectrum is approximately

Fig. 2.1 Neutron spectrum of a thermal reactor

2 Nuclear Reactor Calculations 51

proportional to the fission spectrum χ(E). In the energy region below several

hundred of keV, the fast neutrons from fission lose their energies mainly through

elastic scattering reaction with light moderator nuclides such as hydrogen.

According to neutron slowing-down theory [1], when fast neutron sources are in

an infinite homogeneous medium which is an ideal moderator with negligible

absorption, the neutron energy spectrum behaves as 1/E.

ð2:6Þ

In the practical medium of fuel and moderator, the 1/E distribution is character-

ized by the occurrence of fairly sharp depressions due to the resonance absorption

of 238U, etc. as shown in Fig. 2.1. Moreover, if the resonance absorption is large, the

neutron flux becomes somewhat smaller than the 1/E distribution in the low energy

region.

If neutrons are moderated below several eV of kinetic energy, the kinetic energy

by thermal vibration of nuclei cannot be ignored. In other words, if the kinetic

energies of neutrons become appropriately small, the neutrons collide with ther-

mally vibrating moderator nuclei and their kinetic energies become reversely large.

This is called up-scattering against moderation (down-scattering). In this energy

region, the neutron spectrum is characterized in a thermal equilibrium at a temper-

ature T by a balance between down-scattering and up-scattering. Further, in the

ideal infinite medium without absorption, the thermal equilibrium neutron spectrum

is described by the following Maxwellian distribution function

ð2:7Þ

where k is the Boltzmann constant. At room temperature (T ¼ 300 K), ϕM is

maximized at E ¼ 0.0253 eV for which the corresponding velocity is 2,200 m/s.

On being absorbed, the thermal neutron spectrum deviates a little from ϕM (E)forward the high energy region because absorption cross sections are larger in lower

energies. This is called absorption hardening. To compensate the Maxwellian

distribution for the absorption hardening, neutron temperature, which is a little

higher than moderator temperature, is used as T of Eq. (2.7).

2.1.3 Nuclear Data and Cross Sections

[1] Macroscopic cross sections

The cross section used in the neutron transport equation, Σ, is called the

macroscopic cross section and its unit is (cm�1). The macroscopic cross section

of nuclear reaction x is the sum of the product of atomic number density Ni and

microscopic cross section σix of a constituent nuclide i of the material at a

position~r. It is expressed as Eq. (2.8).


ð2:8Þ

The atomic number density Ni can be given by

ð2:9Þ

where ρi is the density occupied by i in the material,Mi is the atomic mass, and

NA is Avogadro’s number. Variation of atomic number density with position

and time should be considered in power reactors, reflecting fuel burnup or

coolant void fraction.

[2] Microscopic cross sections

Interactions between neutrons and nuclei in nuclear reactors can be classified

broadly into scattering and absorption reactions as shown in Table 2.1.

Scattering is further classified into elastic scattering in which the kinetic energy

is conserved before and after the reaction, and inelastic scattering in which a

part of the kinetic energy is used in exciting a target nucleus. In absorption, the

main reactions are capture, fission, charged-particle emission, and neutron

emission. Thus, the microscopic cross sections of the total, scattering, and

absorption reactions are given by

ð2:10Þ

ð2:11Þ

ð2:12Þ

It is useful to discuss a general energy dependence of these microscopic cross

sections. The neutron energy range to be considered in design of nuclear

reactors is from the Maxwellian distribution of the thermal neutrons at room

temperature to the fission spectrum of the prompt neutrons. Most nuclear design

codes handle the range of 10�5 eV–10 MeV. In this energy range, the

Table 2.1 Classification of key nuclear reactions in nuclear reactors

Classification Reaction Transcription

Cross section

symbol

Reaction

example

Scattering (σs) Elastic scattering (n, n) σe 1H(n, n)

Inelastic scattering (n, n0) σin 238U(n, n0)Absorption

(σa)Radiative capture (n, γ) σγ 238U(n, γ)Fission (n, f ) σf 235U(n, f )

Charged-particle

emission

(n, p) σp 14N(n, p)

(n, α) σα 10B(n, α)Neutron emission (n, 2n)a σ(n, 2n) 9Be(n, 2n)

aThe (n, 2n) reaction can be treated as a special scattering reaction. Here this reaction, which

transmutes the nuclide, is classified as an absorption reaction


microscopic cross sections introduced in Eqs. (2.10), (2.11), and (2.12) behave

as shown in Fig. 2.2.

The elastic scattering cross section is mostly constant in all the energies

except for the MeV region. Meanwhile, in inelastic scattering, the incident

neutron should have sufficient kinetic energy to place the target nucleus in its

excited state. Hence, the inelastic scattering cross section is zero up to some

threshold energy of several MeV. Fast neutrons can be moderated by inelastic

scattering with heavy nuclides, but by elastic scattering with light nuclides

below threshold energies of the heavy nuclides.

Most absorption cross sections including the fission cross section appear as a

straight line with a slope of �1/2 on a log–log scale. This means that the

absorption cross sections are inversely proportional to the neutron speed (1/υlaw) and therefore increase as the neutron energy decreases. Using such large

fission cross sections at low neutron energies and thermal neutrons in the

Maxwellian distribution make it possible that natural or low-enrichment ura-

nium fueled reactors reach a critical state. The current thermal reactors,

represented by LWRs, use the characteristics of the cross section.

For heavy nuclides such as fuel materials, many resonances are observed in

elastic scattering and absorption cross section as shown in Fig. 2.3. The widths

of the resonances broaden as fuel temperature increases. This is called the

Doppler effect. The width broadening facilitates resonance absorption of neu-

trons under moderation. Most low-enrichment uranium fuel is composed of

Fig. 2.2 Energy dependence of cross sections


fertile 238U and thermal neutrons escaping from the resonance of capture

reaction induce fissions for the next generation.

Hence, a rise in fuel temperature leads to a decrease in resonance escape

probability of moderated neutrons and then fission events in the reactor

decrease with thermal neutrons. Such a mechanism is called negative temper-ature feedback. The temperature dependence is not described in the Boltzmann

equation of Eq. (2.1), but reflected in the cross sections of the equation.

[3] Evaluated nuclear data file

The primal data (microscopic cross sections, etc.) for the nuclear reactor

calculation are stored in the evaluated nuclear data file which includes cross

sections uniquely-determined in the whole range of neutron energy on the basis

of fragmentally measured and/or theoretically calculated parameters. The cross

sections are given for all possible nuclear reactions of more than 400 nuclides.

JENDL [2], ENDF/B [3], and JEFF [4] are representative evaluated nuclear

data files.

Extensive data are stored in 80-column text format in the evaluated nuclear

data file as shown in Fig. 2.4 and complicated data such as resonance cross

sections are also compactly-represented by mathematical formulas and their

parameters.

The evaluated nuclear data, from anywhere in the world, are described based

on the same format which might not make sense to biginners. The format has

been changed according to advances in nuclear data and the current latest

format is called ENDF-6 [5]. Various data of delayed neutron fractions, fission

yields, half-lives, etc. as well as cross sections are available.

[4] Multi-group cross sections

The cross sections stored in the evaluated nuclear data file are continuous-

energy data, which are converted to multi-group cross sections through energy

discretization in formats suitable for most of nuclear design calculation codes,

as shown in Fig. 2.5. The number of energy groups (N ) is dependent on nuclear

design codes and is generally within 50–200.

Multi-group cross sections are defined that integral reaction rates for reac-

tions (x ¼ s, a, f,···) of a target nuclide i) should be conserved within the range

Fig. 2.3 Resonance absorption cross section and Doppler effect


of energy group g (ΔEg), as shown in Eq. (2.13). The neutron flux and cross

section in group g are defined as Eqs. (2.14) and (2.15), respectively.

ð2:13Þ

ð2:14Þ

ð2:15Þ

Fig. 2.4 Example of 235U cross sections stored in JENDL (Right bottom figure: plot of the crosssections)

Fig. 2.5 Multi-group cross section by energy discretization


The neutron spectrum of Fig. 2.1 can be used as ϕ(E) in Eq. (2.15) for

thermal reactors. However, while those multi-group cross sections are intro-

duced to calculate neutron flux in reactors, it is apparent that the neutron flux is

necessary to define the multi-group cross sections. This is recursive and con-

tradictory. For example, the depression of neutron flux due to resonance, shown

in the 1/E region of Fig. 2.1, depends on fuel composition (concentrations of

resonance nuclides) and temperature (the Doppler effect). It is hard to accu-

rately estimate the neutron flux before design calculation. In actual practice,

consideration is made for a representative neutron spectrum (ϕw (E): called the“weighting spectrum”) which is not affected by the resonance causing the

depression and is also independent of position. Applying it to Eq. (2.15) gives

Eq. (2.16).

ð2:16Þ

Since these microscopic cross sections are not concerned with detailed

design specifications of fuel and operating conditions of the reactor, it is

possible to prepare them beforehand using the evaluated nuclear data file.

The depression or distortion of neutron flux does not occur for resonance

nuclides at sufficiently dilute concentrations and this is called the infinite

dilution cross section (σi1;x;g). On the other hand, the cross section prepared

in Eq. (2.15) compensates for the neutron spectrum depression due to resonance

in the 1/E region by some method, and is called the “effective (microscopic)

cross section” (σieff ;x;g). There are several approximation methods for the effec-

tive resonance cross section: NR approximation, WR approximation, NRIM

approximation, and IR approximation. For more details, references [1, 6] on

reactor physics should be consulted. The relationship between the effective

cross section and infinite dilution cross section is simply discussed here.

Since neutron mean free path is long in the resonance energy region, a

homogenized mixture of fuel and moderator nuclides, ignoring detailed struc-

ture of the materials, can be considered. If a target nuclide (i) in the mixture has

a resonance, it leads to a depression in the neutron flux at the corresponding

resonance energy as shown in Fig. 2.6. The reaction rate in the energy group

including the resonance becomes smaller than that in the case of no flux

depression (infinite dilution). Hence, the effective cross section is generally

smaller than its infinite dilution cross section. The following definition is the

ratio of the effective cross section to the infinite dilution cross section and

called the self-shielding factor.

ð2:17Þ


In Fig. 2.6, the neutron flux depression becomes large as the macroscopic

cross section (Niσi) of the target nuclide becomes large. On the other hand, if

the macroscopic total cross section ( ) of other nuclides (assuming that

they have no resonance at the same energy) becomes large, the resonance cross

section of the target nuclide is buried in the total cross section of other nuclides

and then the flux depression finally disappears when the total cross section is

large enough. To treat this effect quantitatively, a virtual cross section (the

background cross section) of the target nuclide, converted from the macro-

scopic total cross section, is defined as Eq. (2.18).

ð2:18Þ

The self-shielding factor is dependent on the background cross section and

the temperature of resonance material because the resonance width varies with

the Doppler effect. In fact, the background cross section and material temper-

ature are unknown before design specifications and operating conditions of a

reactor are determined. Hence, self-shielding factors are calculated in advance

and tabulated at combinations of representative background cross sections and

temperatures. Figure 2.7 shows a part of the self-shielding factor table ( f table)for the capture cross section of 238U. The self-shielding factor in design

calculation is interpolated from the f table at a practical background cross

section (σ0) and temperature (T ). The effective microscopic cross section for

neutron transport calculation can be obtained from the self-shielding factor as

ð2:19Þ

Fig. 2.6 Neutron flux

depression in resonance


where T0 is normally 300 K as a reference temperature at which the multi-group

infinite dilution cross sections are prepared from the evaluated nuclear data file.

Recent remarkable advances in computers have made it possible that

continuous-energy data or their equivalent ultra-fine-groups (groups of several

tens of thousands to several millions) are employed in solving the neutron

slowing-down equation and the effective cross sections of resonance nuclides

are directly calculated based on Eq. (2.15). Some open codes, SRAC [7] (using

the PEACO option) and SLAROM-UF [8] for fast reactors, adopt this method.

This method can result in a high accuracy for treating even interference effects

of multiple resonances which cannot be considered by the interpolation method

of a self-shielding factor table, though a long calculation time is required for a

complicated system.

[5] Nuclear data processing and reactor constant library

Nuclear design codes do not use all the massive information contained in the

evaluated nuclear data file and also cannot always effectively read out neces-

sary data because of the data format putting weight on the compact storage.

Therefore, nuclear design codes do not directly read out the evaluated nuclear

data file, but (i) extract necessary data and (ii) prepare a preprocessed data set

(reactor constant library) which is one processed suitably for each code.

A series of tasks is done in order, for example, to select nuclides and cross

section data required for nuclear design, to prepare their infinite dilution cross

sections and self-shielding factor tables for a specified energy group structure of

a nuclear design code, and to store the data in a quickly accessible form.

As nuclear data processing code systems to perform such work, NJOY [9] of

LANL and PREPRO [10] of IAEA have been used all over the world.

Fig. 2.7 Self-shielding factor of 238U capture cross section (ΔEg ¼ 6.48 ~ 8.32 eV)


There are two different forms of the reactor constant library; a continuous-

energy form and a multi-group energy form. The former is used in continuous-

energy Monte Carlo codes such as MCNP [11] and MVP [12] and the latter in

nuclear design codes.

Since most nuclear design codes are provided as a set with a multi-group

reactor constant library, it is usually unnecessary to process the evaluated

nuclear data file in the nuclear design. However, when introducing the latest

evaluated nuclear data file or changing the energy group structure to meet

advances in fuel and design specification of nuclear reactors, a new reactor

constant library is prepared using the nuclear data processing code system. The

reactor constant library, with which a nuclear design calculation begins, might

be regarded as a general-purpose library. However, it should be noted that a

weighting spectrum specified in part for a reactor type and resonance approx-

imations are employed in preparing a multi-group form library.

Nuclear characteristic calculation methods depend on the type of reactor

being targeted. The following discussions center on LWR (especially BWR)

calculation methods which need the most considerations such as heterogeneity

of the fuel assembly, effects of the nuclear and thermal-hydraulic coupled core

calculation, and so on.

2.1.4 Lattice Calculation

[1] Purpose of lattice calculation

Even using a high performance computer, a direct core calculation with several

tens of thousands of fuel pins is difficult to perform in its heterogeneous

geometry model form, using fine-groups (e.g., 107 groups in SRAC) of a

prepared reactor constant library. The Monte Carlo method can handle such a

core calculation, but it is not easy to obtain enough accuracy for a local

calculation or small reactivity because of accompanying statistical errors.

Hence, the Monte Carlo method is not employed for nuclear design calculations

requiring a fast calculation time. Instead, the nuclear design calculation is

performed in two steps: lattice calculation in a two-dimensional (2D) infinite

arrangement of fuel rods or assemblies and core calculation in a three-

dimensional (3D) whole core. The lattice calculation prepares few-group

homogenized cross sections which maintain the important energy dependence

(neutron spectrum) of nuclear reactions, as shown in Fig. 2.8, and this reduces

the core calculation cost in terms of time and memory. Since final design

parameters in the core calculation are not concerned with the energy depen-

dence, the spatial dependence such as for the power distribution is important.

[2] Multi-group neutron transport equation

The neutron transport equation in the lattice calculation is a steady-state

equation without the time differential term in Eq. (2.1). Further, the neutron

energy variable is discretized in the equation and therefore a multi-group form


is used in design codes as shown in Eq. (2.20). The neutron source of Eq. (2.21)

is the multi-group form without the external neutron source of Eq. (2.2) at the

critical condition.

ð2:20Þ

ð2:21Þ

The system to which the multi-group transport equation is applied is an infinite

lattice system of a 2D fuel assembly (including assembly gap) with a reflective

boundary condition. For a complicated geometry, two lattice calculations

corresponding to a single fuel rod and a fuel assembly are often combined.

In practically solving Eq. (2.20) in the lattice model, the space variable (~r) isalso discretized in the equation and each material region is divided into several

sub-regions where neutron flux is regarded to be flat. In liquid metal-cooled fast

reactors (LMFRs), neutron flux in each energy group has an almost flat spatial

distribution within the fuel assembly because the mean free path of the fast

neutrons is long. A simple hexagonal lattice model covering a single fuel rod or

its equivalent cylindrical model simplified to one dimension is used in the

Fig. 2.8 Lattice calculation flow


design calculation of LMFRs. The spatial division can also be simplified by

assigning the macroscopic cross section by material.

On the other hand, thermal reactors have a highly non-uniform distribution

(called the spatial self-shielding effect) of neutron flux in a fuel assembly as

thermal neutron flux rises in the moderator region or steeply falls in the fuel and

absorber as shown in Fig. 2.9. Moreover, control rod guide tubes or water rods

are situated within fuel assemblies and differently enriched fuels or burnable

poison (Gd2O3) fuels are loaded. In such a lattice calculation, therefore, it is

necessary to make an appropriate spatial division in the input data predicting

spatial distribution of thermal neutron flux and its changes with burnup.

Numerical methods of Eq. (2.20) include the collision probability method

(CPM), the current coupling collision probability (CCCP) method, and the

method of characteristics (MOC) [13]. The SRAC code adopts the collision

probability method and can treat the geometrical models as shown in Fig. 2.10.

The collision probability method has been widely used in the lattice calculation,

but it has a disadvantage that a large number of spatial regions considerably

raise the computing cost. The current coupling collision probability method

applies the collision probability method to the inside of fuel rod lattices

constituting a fuel assembly and combines neighboring fuel rod lattices by

neutron currents entering and leaving the lattices. This approach can substan-

tially reduce the assembly calculation cost. Since the method of characteristics

solves the neutron transport equation along neutron tracks, it provides compu-

tations at relatively low cost even for complicated geometrical shapes and it has

become the mainstream in the recent assembly calculation [14].

Fig. 2.9 Example of spatial

division in rectangular

lattice model of LWRs


In lattice calculation codes, effective microscopic cross sections are first

prepared from fine-group infinite dilution cross sections based on input data

such as material compositions, dimensions, temperatures, and so on. The effec-

tive cross sections are provided in solving Eqs. (2.20) and (2.21) by the use of

the collision probability method, etc. and then multi-group neutron spectra are

obtained in each divided region (neutron spectrum calculation).

[3] Homogenization and group collapsing

In the core calculations with a huge amount of space-dependent data (cross

section and neutron flux), the effective cross sections are processed, with a little

degradation in accuracy as possible, by using the results from the multi-group

lattice calculation. There are two processing methods. One is homogenization

to lessen the space-dependent information and the other is group-collapsing to

reduce the energy-dependent information as shown in Fig. 2.11. The funda-

mental idea of both methods is to conserve neutron reaction rate. In the

homogenization, a homogenized neutron flux ϕhomog is first defined as averaged

flux weighted by volume Vk of region (k). Next, a homogenized cross section

Σhomox;g is determined as satisfying Eq. (2.23) to represent the reaction rate in the

homogenized whole region of volume Vhomo. The fine-group neutron flux ϕg,k is

used to calculate the homogenized cross section in Eq. (2.24).

ð2:22Þ

ð2:23Þ

Fig. 2.10 Lattice models of SRAC [7]


ð2:24Þ

Similarly, the homogenized microscopic cross section σi;homox;g which repre-

sents the homogenized region is defined. First, the homogenized atomic number

density Ni,homo of nuclide i is defined from the following conservation of

number of atoms

ð2:25Þ

The homogenized microscopic cross section is then given from the conser-

vation of microscopic reaction rate of nuclide i as

ð2:26Þ

Such homogenized neutron flux ϕhomog and cross section ∑ homo

x;g (or σi;homox;g )

terms are used in performing the energy-group collapse. The fine-groups (g) arefirst apportioned to few-groups (G) for the core calculation as shown in

Fig. 2.11. Since the multi-group neutron flux was obtained by the integration

over its energy group, the few-group homogenized neutron flux can be

represented by

Fig. 2.11 Homogenization and group collapsing of cross section


ð2:27Þ

The next step is to consider the conservation of reaction rate in energy group

G in the same manner as that in the homogenization.

The few-group homogenized macroscopic and microscopic cross sections

are then given by

ð2:28Þ

ð2:29Þ

The number of few-groups depends on reactor type and computation code.

Two or three groups are adopted for the nuclear and thermal-hydraulic coupled

core calculation of LWRs and about 18 groups are used for the core calculation

of LMFRs. It should be noted that the conservation of reaction rate has been

considered under the assumption that the lattice system can be represented as an

infinite array of identical lattice cells. If the neutron spectrum in the actual core

system is very different from the multi-group neutron spectrum calculated in

the infinite lattice system, the applicability of such few-group homogenized

cross sections to the core calculation is deteriorated. In this case, it is effective

to increase the number of energy groups in the core calculation, but that gives a

more costly computation.

[4] Lattice burnup calculation

The lattice burnup calculation prepares few-group homogenized cross sections

with burnup on the infinite lattice system of fuel assembly. The series of lattice

calculation procedures described in Fig. 2.8 are repeatedly done considering

changes in fuel composition with burnup.

However, the lattice burnup calculation is not carried out in the design

calculation of LMFRs which have a high homogeneity compared with LWRs

and do not lead to a large change in neutron spectrum during burnup. Hence a

macroscopic cross section at a position (~r) in the reactor core can be formed by

Eq. (2.30) using the homogenized microscopic cross section prepared in the

lattice calculation. At this time, the homogenized atomic number densities are

calculated according to burnup of each region during the reactor core

calculation.

ð2:30Þ

By contrast, when considering 235U in a fuel assembly of a LWR as an

example, the constituents of UO2 fuel, MOX fuel, or burnable poison fuel

(UO2-Gd2O3) each lead to different effective cross sections and composition

variations with burnup. It is difficult to provide common few-group


homogenized microscopic cross sections and homogenized atomic number

densities suitable for all of them. Moreover, space-dependent lattice calcula-

tions by a fuel assembly model during the core calculation require an enormous

computing time. In the design of reactors with this issue of fuel assembly

homogenization, therefore, the lattice burnup calculation is performed in

advance for the fuel assembly type of the core and then few-group homoge-

nized macroscopic cross sections are tabulated with respect to burnup, etc. to be

used for the core calculation. In the core calculation, macroscopic cross sec-

tions corresponding to burnup distribution in the core are prepared by interpo-

lation of the table (the table interpolation method of macroscopic cross

sections) [15].

Atomic number densities of heavy metals and fission products (FPs) are

calculated along a burnup pathway as shown in Fig. 2.12, called the burnupchain.

Fig. 2.12 An example burnup chain of heavy metals [7]


The time variation in atomic number density (Ni) of a target nuclide (i) canbe expressed as the following differential equation.

ð2:31Þ

where the first term on the right-hand side expresses the production rate of

nuclide i due to radioactive decay of another nuclide j on the burnup chain. λj isthe decay constant of nuclide j and f j!i is the probability (branching ratio) of

decay to nuclide i. The second term is the production rate of nuclide i due to thenuclear reaction x of another nuclide k. The major nuclear reaction is the

neutron capture reaction and other reactions such as the (n,2n) reaction can

be considered by necessity. hσkxϕi, which is the microscopic reaction rate of

nuclide k integrated over all neutron energies, is calculated from the fine-group

effective cross section and neutron flux in the lattice calculation. gk!ix is the

probability of transmutation into nuclide i for nuclear reaction x of nuclide k.The third term is for the production of FPs. Fl is the fission rate of heavy nuclide

l and γl ! i is the production probability (yield fraction) of nuclide i for thefission reaction. The last term is the loss rate of nuclide i due to radioactive

decay and absorption reactions.

Applying Eq. (2.31) to all nuclides including FPs on the burnup chain gives

simultaneous differential equations (burnup equations) corresponding to the

number of nuclides. Ways to solve the burnup equations include the Bateman

method [16] and the matrix exponential method [17]. This burnup calculation,

called the depletion calculation, is performed for each burnup region in case of

multiple burnup regions like a fuel assembly and it gives the variation in

material composition with burnup in each region. The lattice calculation is

carried out with the material composition repeatedly until reaching maximum

burnup expected in the core.

The infinite multiplication factor calculated during the lattice calculation is

described as the ratio between neutron production and absorption reactions in

an infinite lattice system without neutron leakage.

ð2:32ÞFigure 2.13 shows the infinite multiplication factor obtained from the lattice

burnup calculation of a BWR fuel assembly. Since thermal reactors rapidly

produce highly neutron absorbing FPs such as 135Xe and 149Sm in the beginning

of burnup until their concentrations reach the equilibrium values, the infinite


multiplication factor sharply drops during a short time. Then the infinite

multiplication factor monotonously decreases in the case of no burnable poison

fuel (UO2 + Gd2O3). Meanwhile, it increases once in burnable poison fuel

because burnable poisons burn out gradually with burnup, and then it decreases

when the burnable poisons become ineffective.

The few-group homogenized macroscopic cross sections are tabulated at

representative burnup steps (E1, E2, E3, . . .) by the lattice burnup calculation

and stored as a reactor constant library for the core calculation, as shown in

Fig. 2.14. Three different moderator densities (ρ 1, ρ 2, ρ 3) for one fuel type are

assumed in the lattice burnup calculation of BWRs. Moderator density of BWRs

Fig. 2.13 Lattice burnup calculation of fuel assembly

1ρ2ρ

3ρ

E1 E2 E3 E4¼

k¥

1ρ

2ρ

3ρ

E1 E2 E3¼

Burnup

Burnup (Exposure)

Histo

rica

l m

oder

ator

den

sity

Two-dimensional table of S x,G

Fig. 2.14 Tabulation of few-group homogenized macroscopic cross sections in lattice burnup

calculations


is provided as a function of void fraction α for two densities ρl and ρg in liquid

and vapor phases, respectively, in the steam table. That is, it is given by

ð2:33Þ

where ρ 1, ρ 2, and ρ 3 are the moderator densities corresponding to each void

fraction at the core outlet (α ¼ 0.7), as an average (α ¼ 0.4), and at the inlet

(α ¼ 0.0) of typical BWRs. Since this moderator density is assumed to be

constant as an average value during each lattice burnup step, it is called a

historical moderator density.The few-group homogenized macroscopic cross sections are tabulated

according to two historical parameters (burnup E and historical moderator

density ρ ) by energy group (G) or cross section type (x). The cross section tablesare prepared individually according to fuel type (F), control rod insertion or

withdrawal, etc. Furthermore, homogenized microscopic cross sections and

atomic number densities of 135Xe, 149Sm, 10B, etc. can be tabulated by necessity.

[5] Branch-off calculation

In the lattice burnup calculation, a combination of parameters such as moderator

density and temperature, and fuel temperature is made at representative values

ðρ 0, Tm0, Tf0) expected at normal operation of the reactor. In the core calculation,

however, a different set (ρ, Tm, Tf) from the representative set is taken depending

on position and time. Hence, a subsequent calculation called the branch-offcalculation is performed after the lattice burnup calculation if necessary.

Figure 2.15 depicts an example branch-off calculation of moderator density.

The calculation proceeds in the following order:

(i) Perform the lattice burnup calculation at a reference condition (ρ 0, Tm0,Tf0). Designate the few-group homogenized cross section prepared from

the lattice burnup calculation as Σ (ρ 0, Tm0, Tf0).

aρ

0ρbρ

E1 E2 E3 E4Burnup

Fig. 2.15 Branch-off

calculation of moderator

density


(ii) Perform the lattice calculation at each burnup step on the condition that

only the moderator density is changed from ρ 0 to ρa, by using the same

fuel composition at each burnup step. Designate the few-group homoge-

nized cross section prepared from the lattice calculation (or the branch-off

calculation) as Σ(ρ 0 ! ρa, Tm0, Tf0).(iii) Carry out the branch-off calculation similar to (ii) at another moderator

density of ρb and give Σ(ρ 0 ! ρb, Tm0, Tf0).(iv) If the moderator density is instantaneously changed fromρ 0 to an arbitrary

ρ, calculate the corresponding cross section by the following approxima-

tion (quadratic fitting).

ð2:34Þ

(v) Determine the fitting coefficients a and b from the approximation at

ρ ¼ ρa and ρ ¼ ρb.

Hence, the cross section at an arbitrary moderator density (an instantaneous

moderator density) ρ away from the historical moderator density ρ 0 can be

expressed by the method mentioned above. If three moderator densities (ρ 1, ρ 2,

ρ 3) are employed as ρ 0 as shown in Fig. 2.14, their branch-off calculations can

give Σ (ρ 1 ! ρ, Tm0, Tf0), Σ (ρ 2 ! ρ, Tm0, Tf0), and Σ (ρ 3 ! ρ, Tm0, Tf0). Byinterpolation of the three points, the cross section at ρ resulting from an

instantaneous change from a historical moderator density ρ can be obtained

as Σ (ρ ! ρ, Tm0, Tf0).The cross section at an instantaneous change in fuel or moderator tempera-

ture can also be described by the same function fitting as above. Since its

change in cross section is not as large as a void fraction change (0–70 %), the

following linear fittings are often used.

ð2:35Þ

ð2:36Þ

Equation (2.36) has square roots of fuel temperature in order to express the

cross section change due to the Doppler effect by a lower-order fitting equation.

An employment of a higher-order fitting equation can lead to a higher expres-

sion capability and accuracy, but it gives rise to a substantial increase in the

number of branch-off calculations and fitting coefficients and therefore results

in an inefficient calculation. Thus, the reference cross sections and their fitting

coefficients are stored into the few-group reactor constant library and used in

the nuclear and thermal-hydraulic coupled core calculation.


2.1.5 Core Calculation

[1] Diffusion equation

In core calculations such as the nuclear and thermal-hydraulic coupled core

calculation, core burnup calculation, or space-dependent kinetics calculation,

high-speed calculation is the biggest need. Especially since large commercial

reactors have a considerably larger core size compared with critical assemblies

or experimental reactors, the neutron transport equation (2.1) has to take many

unknowns and hence need a high computing cost.

For most core calculations, the angular dependence (~Ω) of neutron flux is

disregarded and the neutron transport equation is approximately simplified as

Eq. (2.37).

ð2:37Þ

Its physical meaning is simple. As an example, a space of unit volume

(1 cm3) like one die of a pair of dice is considered. The time variation in

neutron population within the space [the LHS of Eq. (2.37)] is then given by the

balanced relationship between neutron production rate (S), net neutron leakage

rate through the surface (∇�~J), and neutron loss rate by absorption (Σaϕ).~J is referred to as the net neutron current and expressed by the following

physical relation known as Fick’s law

ð2:38Þ

whereD is called the diffusion coefficient; it is tabulated together with few-groupcross sections in the lattice calculation to be delivered to the core calculation.

Inserting Eq. (2.38) into Eq. (2.37) gives the time-dependent diffusion equation.

ð2:39Þ

The time-independent form of Eq. (2.39) is the steady-state diffusion equa-

tion and all types of the core calculation to be mentioned thereafter are based on

the equation

ð2:40ÞThus, solving the diffusion equation in critical reactors is to regard the reactor

core as an integration of subspaces like the dice cubes and then to find a neutron

flux distribution to satisfy the neutron balance between production and loss in all

the spaces (corresponding to mesh spaces to be described later).


[2] Solution to one-dimensional (1D) one-group diffusion equation by finite

difference method

First a non-multiplying medium (e.g., water) is considered for simplicity of the

equation, though practical systems are composed of various materials with

different cross sections. In addition, it is assumed that all neutrons can be

characterized by a single energy (one-group problem). Therefore, the cross

section does not depend on location and in this case the diffusion equation is

given by Eq. (2.41).

ð2:41Þ∇2 (the Laplacian operator) is dependent on the coordinate system and in the

Cartesian coordinate system it is represented as Eq. (2.42).

ð2:42Þ

In an infinitely wide medium in y and z directions, the diffusion equation

reduces to the 1D form.

ð2:43Þ

Here a uniform neutron source (s1 ¼ s2 ¼ s3 ··· ¼ s) in a finite slab of width2d is considered as shown in Fig. 2.16 and the finite difference method to solve

for neutron flux distribution is discussed. The reflective boundary condition is

applied to the center of the slab and one side of width d is divided into equal

N regions (spatial meshes). The second-order derivative of Eq. (2.43) can be

Fig. 2.16 Finite difference in 1D plane geometry


approximated1 by using neutron flux ϕi at a mesh boundary i and neutron fluxesat both adjacent sides as follows;

ð2:44Þ

Hence the diffusion equation at each mesh point can be given by

ð2:45Þ

It can be rewritten by

ð2:46Þwhere

ð2:47Þ

In the case of N ¼ 5, the following simultaneous equations are gotten.

ð2:48Þ

Since there are six unknowns (ϕ0 to ϕ5) among four equations in Eq. (2.48),

another two conditions are needed to solve the simultaneous equations. Bound-

ary conditions at both ends (x ¼ 0 and x ¼ d ) of the slab are given by the

following.

Vacuum boundary condition (an extrapolated distance is assumed zero):

ð2:49Þ

1Most practical codes of the finite difference method take neutron fluxes at the center points of

divided meshes [13]. A simple and easy to understand method was employed here.


Reflective boundary condition:

ð2:50Þ

Equation (2.51) is obtained by inserting Eqs. (2.49) and (2.50) into Eq. (2.48)

and rewriting in matrix form.

ð2:51Þ

A 1D problem generally results in a tridiagonal matrix equation of (N � 1)

� (N � 1) systemwhich can be directly solved using numerical methods such as

the Gaussian elimination method. Since the finite difference method introduces

the approximation as Eq. (2.44), it is necessary to choose a sufficiently small mesh

spacing. The spatial variation of neutron flux is essentially characterized by the

extent of neutron diffusion such as the neutronmean free path. Hence the effect of

themesh size on nuclear characteristic evaluations of a target reactor coremust be

understood and then the mesh size is optimized relative to the computation cost.

[3] Solution of the multi-dimensional diffusion equation

The diffusion equation in a 2D homogeneous plane geometry is given by

Eq. (2.52).

ð2:52Þ

The plane is divided into N and M meshes in x and y directions, respectively(Fig. 2.17). Similarly to the 1D problem, the second-order derivatives can be

Fig. 2.17 Finite difference

in 2D plane geometry


approximated by using neutron fluxes at a point (i, j) and its four adjacent onesas the following.

ð2:53Þ

ð2:54Þ

Hence the diffusion equation at each mesh point can be given by

ð2:55Þ

and it can be rewritten as Eq. (2.56).

ð2:56ÞNext, the corresponding simultaneous equations are expressed in matrix

form, similarly to the 1D problem except four boundary conditions are used:

two conditions in each of the x and y directions.A 3D problem even in a different coordinate system basically follows the

same procedure and a matrix form of ([A]ϕ ¼ S) can be taken as shown in

Fig. 2.18.

Fig. 2.18 Matrix equation forms of finite difference method


In the case of a large-size matrix [A], a direct method such as the Gaussian

elimination leads to increased necessary memory size and it is liable to cause

accumulation of numerical errors as well. Hence, generally first the matrix [A]is composed into its diagonal element [D] and off-diagonal element [B]. Then ϕ

is guessed and new guesses are calculated iteratively to converge to the true

solution (Fig. 2.19). This iterative algorithm, such as found in the Gauss-Seidel

method or the successive overrelaxation (SOR) method, is frequently referred

to as the inner iteration.[4] Solution to multi-group diffusion equation

As already mentioned, the diffusion equation is a balanced equation between

neutron production and loss. The multi-group diffusion theory is similarly

considered except that the neutron energy is discretized into multi-groups. As

shown in Fig. 2.20, subsequent treatment should be made for neutrons moving

[A]=[D] + [B]

[A]φ = S

= +

φ =[D]-1S-[D]-1[B] φ

φconverged?

initial guess

update φ

inneriteration

convergence

end

Fig. 2.19 Inner iteration to solve a system of equations

g = 1

Group 1 Group 2 Group 3

g = 2

g = 3

Fas

tG

roup

sT

herm

alG

roup

s 4 eV

800 keV

10 MeV

10-5 eV

E

S1

S2

S3

Fig. 2.20 Slowing-down neutron balance in three-group problem


to other groups through slowing down which is not considered in the one-group

diffusion theory. In particular, there are two points:

(i) Neutrons which move out of a target group to other groups by slowing

down result in neutron loss in the neutron balance of the target group.

(ii) Neutrons which move in a target group from other groups by slowing

down result in neutron gain in the neutron balance of the target group.

Hence, the multi-group diffusion equation introduces the cross section Σg!g0

(scattering matrix) characterizing neutrons which transfer from group g to

group g0 by elastic or inelastic scattering.

In the example of the three-group problem presented in Fig. 2.20, the neutron

balance equation of each group can be given by

ð2:57Þ

ð2:58Þ

ð2:59Þ

It is seen that neutron loss due to slowing down in a group results in neutron

gains in other groups. Σ1!2ϕ1 in Eq. (2.58) and Σ1!3ϕ1 + Σ2!3ϕ2 in Eq. (2.59)

are referred to as the “slowing-down neutron source” since they represent the

neutron source due to slowing down from other groups.

Equation (2.57) for group 1 can be rewritten as the following.

ð2:60Þ

ð2:61ÞThe sum of absorption cross section and group-transfer cross section,

namely, the cross section which characterizes neutrons removed from the target

group, is called the removal cross section. It is observed that Eq. (2.60)

rewritten using the removal cross section is in the same form as the

one-group diffusion equation and can be solved by the finite difference method

mentioned before. Substituting the obtained ϕ1 for the one in the slowing-down

source of Eq. (2.58) makes it possible to calculate the neutron flux of group

2, ϕ2, in the same way. Further, the neutron flux of group 3, ϕ3, can be also

obtained by substituting ϕ1 and ϕ2 into Eq. (2.59).

Thus, the multi-group diffusion equations can be solved in sequence from

the highest energy group. However, the solution is not obtained by this method

if the thermal energy region (up to about 4 eV) is divided into multi-groups.

As an example, a four-group diffusion problem using two fast groups and two

thermal groups is considered.


ð2:62Þ

ð2:63Þ

ð2:64Þ

ð2:65Þ

The fast-group neutron fluxes ϕ1 and ϕ2 can be derived by the same

procedure as in the previous three-group problem. In the thermal groups,

however, an event that a neutron gains kinetic energy in a collision with a

moderator nuclide in thermal vibration, that is, the upscattering of thermal

neutrons should be considered. The last term of Eq. (2.64) represents that.

Thus, since the neutron source terms for group 3 include the unknown neutron

flux of group 4, ϕ4, it is impossible to directly solve the equation. Here, a guess

ϕð0Þ4 instead of ϕ4 is taken at Eq. (2.64) and then a solution ϕð1Þ

3 is given.

Moreover, a solution ϕð1Þ4 is also acquired by the substitution of ϕð1Þ

3 into

Eq. (2.65). Since ϕð1Þ3 and ϕð1Þ

4 are not correct solutions, another new guess

ϕð1Þ4 is used in Eq. (2.64) to obtain ϕð2Þ

3 , which is substituted again into

Eq. (2.65) to acquire ϕð2Þ4 . Such calculations are iteratively performed until

ϕðnÞ3 and ϕðnÞ

4 come to an agreement with each previous ϕðn�1Þ3 and ϕðn�1Þ

4 . This

calculation is referred to as the “thermal iteration” calculation.

The codes developed for core design of only fast reactors may have no such

upscattering and iteration calculation function or may cut back the slowing-

down groups by assuming collisions with only heavy metal elements such as

sodium. Since such codes cannot correctly manage transport calculation of

thermal neutrons, they should not be used carelessly in a thermal reactor

calculation even though the codes solve the same fundamental equation.

[5] Eigenvalue problem with fission source

So far the diffusion equation has been discussed in a non-multiplying medium

with an external neutron source. Here, fission neutron production (fission

source) in a reactor core is applied to the diffusion equation. Assuming a

volumetric fission reaction rate Σf ϕ and an average number of neutrons

released per fission v in a unit cubic volume (1 cm3) leads to a volumetric

fission neutron production rate by multiplying both ones. Hence, substituting

the fission source vΣf ϕ for the external source (S) in Eq. (2.41) gives the

one-group diffusion equation at the steady-state reactor core as the following.


ð2:66Þ

This equation represents a complete balance between the number of neutrons

produced by fission and the number of neutrons lost due to leakage and

absorption, in an arbitrary unit volume at the critical condition of the reactor.

In actual practice, however, it is hard to completely meet such a balance in the

design calculation step. A minor transformation of Eq. (2.66) can be made as

ð2:67Þ

where keff is an inherent constant characterizing the system, called the eigen-value. If keff ¼ 1.0, Eq. (2.67) becomes identical to Eq. (2.66). The equation

may be solved for the eigenvalue keff:

ð2:68Þ

where L, A, and P denote neutron leakage, absorption, and production, respec-

tively. If the space is extended to the whole reactor core (i.e., each term is

integrated over the whole reactor core), keff can be interpreted as follows. The

numerator is the number of neutrons that will be born in the reactor core in the

next generation, whereas the denominator represents those that are lost from

the current generation. Hence, the condition of the reactor core depends on the

keff value as given next.

ð2:69Þ

Another interpretation can be given for keff in Eq. (2.67). As mentioned

above, it is highly unlikely to hit on the exact neutron balance of L + A ¼ P in

a practical core design calculation. Neither the supercritical (L + A < P) norsubcritical (L + A > P) condition gives a steady-state solution to Eq. (2.66).

Hence, if keff is introduced into the equation as an adjustment parameter like

Eq. (2.67), the neutron balance can be forced to maintain (P ! P/keff) and the

equation will always have a steady-state solution. In a supercritical condition

(keff > 1), for example, the neutron balance can be kept by adjusting down the

neutron production term (νΣfϕ ! νΣfϕ/keff).A problem expressed as an equation in the form of Eq. (2.67) is referred to as

an eigenvalue problem. By contrast, a problem with an external neutron source

such as Eq. (2.41) is referred to as a fixed-source problem. The big difference

between both equations is that the eigenvalue problem equation has an infinite

set of solutions. For example, if ϕ is a solution of Eq. (2.67), it is self-evident


that 2ϕ and 3ϕ are also other solutions. Consequently, the solutions of the

eigenvalue problem are not absolute values of ϕ but the eigenvalue keff and a

relative spatial distribution of ϕ. Meanwhile, the fixed-source problem has a

spatial distribution of absolute ϕ which is proportional to the intensity of the

external neutron source.

[6] Solution to multi-group eigenvalue problem

The diffusion equation of the eigenvalue problem by the one-group theory

[Eq. (2.67)] can be extended to an equation based on the multi-group theory

and its numerical solutions are discussed here.

In the eigenvalue problem of the multi-group theory, it is necessary to

describe the fission source term (νΣfϕ) in the multi-group form as well as the

loss and production terms due to the neutron slowing-down. Since the fission

reaction occurs in all energy groups, the total production rate of fission neutrons

in the unit volume is given by Eq. (2.70).

ð2:70Þ

The probability that a fission neutron will be born with an energy in group

g is given by

ð2:71Þ

where χ(E) is the fission neutron spectrum normalized to unity and therefore

the sum of χg is the same unity.

ð2:72Þ

Hence, the fission source of group g can be expressed as Eq. (2.73).

ð2:73Þ

For the three-group problem (fixed-source problem) of Fig. 2.20, replace-

ment of the external source by the fission source is considered. If Sg is

substituted by χgP/keff in Eqs. (2.57), (2.58), and (2.59), then the following

diffusion equations are given for three-group eigenvalue problem:

ð2:74Þ

ð2:75Þ


ð2:76Þ

where

ð2:77Þ

In the three-group fixed-source problem, it was seen that the diffusion

equations are solved in consecutive order from group 1. However,

Eqs. (2.74), (2.75), and (2.76) have the fission source P which includes ϕ1,

ϕ2, and ϕ3, and moreover keff is unknown. Hence, iterative calculations are

performed as the next sequence.

(i) Guess kð0Þeff for keff, and ϕ

ð0Þ1 , ϕð0Þ

2 , and ϕð0Þ3 for determining P. It is common

to begin with an initial guess of kð0Þeff ¼ 1.0 and a flat distribution of the

neutron fluxes (constant values).

(ii) Since neutron fluxes are relative and arbitrary values in eigenvalue

problems, normalize the initial neutron fluxes to satisfy Eq. (2.78).

ð2:78Þ

Since fission source terms are divided by keff in eigenvalue diffusion

equations, Eq. (2.78) indicates that the total neutron source in the system

is normalized to a value of 1 to fit into the next diffusion equation.

(iii) Provide P(0) of Eq. (2.79) with the RHS of Eq. (2.80) and then solve this

for ϕð1Þ1 .

ð2:79Þ

ð2:80Þ

(iv) Substitute the solution of Eq. (2.80), ϕð1Þ1 , for the second term in the RHS

of Eq. (2.81) and then solve this for ϕð1Þ2 .

ð2:81Þ

(v) Substitute again the solutions ϕð1Þ1 and ϕð1Þ

2 from Eqs. (2.80) and (2.81) for

the RHS of Eq. (2.82) and then solve this for ϕð1Þ3 .


ð2:82Þ

(vi) Recalculate keff by ϕð1Þ1 , ϕð1Þ

2 , and ϕð1Þ3 obtained in (iii) to (v) like

Eq. (2.83)

ð2:83Þ

Here, keff is interpreted as the number of neutrons that will be born in the next

generation finally, considering moderation and transport in the whole core,

when assuming that one neutron is given as a fission source to the whole core.

This is the definition of the effective multiplication factor.

Those solutions are not the correct ones yet because they are based on the

initial guesses. The calculations from (iii) to (vi) are iteratively performed until

keff and all of the group fluxes converge. This iterative calculation is known as

the outer iteration calculation. The outer iteration test for convergence is done

by comparing values at an iteration (n) with those at its previous iteration

(n � 1):

ð2:84Þ

ð2:85Þ

where εk and εϕ are the convergence criteria of the effective multiplication

factor and neutron flux, respectively.

The relative neutron flux distribution ϕg ~rð Þ has absolute values due to the

thermal power Q of the reactor. The normalization factor A of ϕg ~rð Þ to an

absolute neutron flux distribution Φg ~rð Þ can be determined by

ð2:86Þ

ð2:87Þ

where κ is the energy released per fission (about 200 MeV). Finally, the absolute

neutron flux distribution for the thermal power can be given as the following.

ð2:88Þ


The distribution of the thermal power within the reactor core is

ð2:89Þ

which is used as a heat source for the thermal-hydraulic calculation.

A general form of the multi-group diffusion equation in the case of the

eigenvalue problem is given by Eq. (2.90), which can be solved in the same

way considering the balance between neutron production and loss in group g asshown in Fig. 2.21. It is seen that Eqs. (2.74), (2.75), and (2.76) are also

represented by the general form.

ð2:90Þ

[7] Nodal diffusion method

The finite difference method is widely used in the design calculation of

fast reactors, the analysis of critical assembly experiments, and so on. For

fast reactors, convergence of the outer iteration is fast due to the long mean free

paths of neutrons, and moreover, the nuclear and thermal-hydraulic coupled

core calculation is not needed. Hence, the computation speed required for the

design calculation can be achieved even with fine meshes in the finite differ-

ence method. For conventional LWRs as shown in Fig. 2.22, however, a fine

meshing of 1–2 cm is necessary to obtain a high-accuracy solution because the

neutron mean free path is short.

For example, a 3D fine meshing of a PWR core of over 30,000 liters leads to

a formidable division of several tens of millions of meshes even though

Fig. 2.21 Balance of neutron flux in group g


excluding the reflector region. Furthermore, the neutron diffusion equation

is repeatedly solved in the nuclear and thermal-hydraulic coupled core calcu-

lation, core burnup calculation, space-dependent kinetics calculation, and so

on. Hence, a calculation using the 3D finite difference method with such a fine

meshing is extremely expensive even on a current high-performance computer

and therefore it is impractical. The nodal diffusion method [18] was, therefore,

developed to enable a high-speed and high-accuracy calculation with a coarse

meshing comparable to a fuel assembly pitch (about 20 cm). It is currently the

mainstream approach among LWR core calculation methods [19].

The numerical solution of the nodal diffusion method is somewhat compli-

cated and is not discussed here. The main features of the approach are briefly

introduced instead.

(i) Since the coarse spatial mesh (node) is as large as a fuel assembly pitch,

the number of unknowns is drastically reduced compared with that of the

finite difference method.

(ii) The 3D diffusion equation in a parallelepiped node (k) is integrated over

all directions except for the target direction and then is reduced to a 1D

diffusion equation including neutron leakages in its transverse directions.

For example, the diffusion equation in the x direction is expressed as

Fig. 2.22 Comparison of conventional LWRs and critical assembly (TCA) in size


ð2:91Þ

where Δky and Δ

kz are the mesh widths in y and z directions, respectively. Lky

(x) and Lkz (x) represent the neutron leakage in each direction. These

unknown functions are provided by a second-order polynomial expansion

using the transverse leakages of two adjacent nodes.

(iii) Typical solutions to the 1D diffusion equation of Eq. (2.91) are (a) the

analytic nodal method [20] for two-group problems, (b) the polynomial

expansion nodal method [18, 21] to expand ϕkx(x) into a about fourth-

order polynomial, and (c) the analytic polynomial nodal method [22, 23]

to expand Skx(x) into a second-order polynomial and to express ϕkx(x) as an

analytic function.

A 3D benchmark problem [24] given by the IAEA as shown in Fig. 2.23 is

taken as a calculation example for the suitability of the nodal diffusion method.

A PWR core model is composed of two types of fuels and control rods are

inserted at five locations of the quadrant inner fuel region. At one location,

control rods are partially inserted to 80 cm depth from the top of the core.

The meshing effect on the power distribution is relatively large in this case and

hence this benchmark problem has been widely employed to verify diffusion

codes.

The calculation results using MOSRA-Light code [21] which is based on the

fourth-order polynomial nodal expansion method are shown in Fig. 2.24 for two

Fig. 2.23 3D benchmark problem of IAEA


mesh sizes (20 and 10 cm). The reference solution has been taken by an extrap-

olation to zero size from the five calculations with different mesh sizes using a

finite difference method code. For the effective multiplication factor, the discrep-

ancy with the reference value is less than 0.006 % Δk in either case and thus the

meshing effect can be almost ignored. For the assembly power, the discrepancy in

the case of 20 cm mesh is as small as 0.6 % on average and 1.3 % at maximum.

The discrepancy becomes smaller than 0.5 % on average in the case of a mesh size

of 10 cm or less. It is noted that the finite difference method requires a mesh size

smaller than 2 cm and more than 100 times longer computation time to achieve the

same accuracy as that in the nodal diffusion method.

Thus the IAEA benchmark calculation indicates the high suitability of the

nodal diffusion method to LWR cores which have fuel assemblies of about 14–

21 cm size. An idea of the nodal diffusion method is its approach to decompose

the reactor core into relatively large nodes and then to determine the neutron

flux distribution within each node to maintain the calculation accuracy. For

example, the polynomial nodal expansion method introduces the weighted

residual method to obtain high-order expansion coefficients. It leads to an

increased number of equations to be solved.

In other words, the high suitability of the nodal diffusion method to practical

LWR calculation results is because the computation cost reduction due to a

substantial decrease in the number of meshes surpasses the cost rise due to an

increase in number of equations. Conversely, if it is possible to reach sufficient

accuracy with the same meshing, the finite difference method will be effective

Fig. 2.24 Comparison of effective multiplication factor and assembly power distribution by nodal

diffusion method [21]


because it is not necessary to solve extra equations. Hence, the nodal diffusion

method does not need to have an advantage over the finite difference method

for the analysis of fast reactors or small reactors.

[8] Nuclear and thermal-hydraulic coupled core calculation

In the nuclear reactor design calculation, the thermal-hydraulic calculation is

performed based on information on the heat generation distribution acquired

from the nuclear calculation of the reactor core. In LWRs, the parameters such

as moderator temperature, moderator density or void fraction, and fuel tem-

perature obtained from the thermal-hydraulic calculation have a large effect on

nuclear characteristics (nuclear and thermal-hydraulic feedback). The nuclear

and thermal-hydraulic calculations should be mutually repeated until parame-

ters of both calculations converge. This coupled calculation shown in Fig. 2.25

is referred to as the nuclear and thermal-hydraulic coupled core calculation

(hereafter the N-TH coupled core calculation). The procedure of the N-TH

coupled core calculation using the macroscopic cross section table prepared

from lattice burnup calculations is discussed for a BWR example.

Two types of parameters are used in the coupled calculation. One is histor-

ical parameters related to the fuel burnup and the other is instantaneous

parameters without a direct relation to it. The historical parameters such as

burnup are obtained from the core burnup calculation discussed in the next

section. Here it is assumed that all of the historical parameters are known. The

macroscopic cross section is first required for the nuclear calculation of the

)](),([ rIrH guess (r, Tm, Tf , NXe …)I =

(E, r, …)H =

Feedbackcross sectioncalculation

Nuclearcalculation

)(rInew)(rΣ

)(),(, rq′′′rkeff φ

)(rTf

Cross-sectiontable

)(),( rNrN SmXe

Thermal-Hydrauliccalculation

)(10 rNB

Fuel temperaturecalculation

Iterativecalculation

Heat transfer calculationor Table interpolation

Historical parameters

Instantaneous parameters

< End >

< Start >

Enthalpy rise Void fractionPressure dropInlet flow rate

r(r), Tm (r), SB (r)

Fig. 2.25 Nuclear and thermal-hydraulic coupled core calculation of LWR


reactor core. The space and time-dependent macroscopic cross section of a

LWR is, for example, represented as Eq. (2.92).

ð2:92Þ

F: fuel type (initial enrichment, Gd concentration, structure type, etc.)

R: control rod type (absorber, number of rods, concentration, structure type, etc.)

E: burnup (GWd/t)ρ : historical moderator density (the burnup-weighted average moderator

density)

ð2:93Þ

ρ: moderator density (also called the instantaneous moderator density against ρ )in which the void fraction α in BWRs is calculated from two densities ρl andρg in liquid and vapor phases, respectively

ð2:94Þ

Tf: fuel temperature (the average fuel temperature in the fuel assembly)

Tm: moderator temperature (the average moderator temperature in the fuel

assembly)

Ni: homogenized atomic number density of nuclide i (e.g., 135Xe or 149Sm) for

which the atomic number density changes independently of the burnup or the

historical moderator density and which has an effect on the core reactivity

depending on the operation condition such as reactor startup or shutdown

SB: concentration of soluble boron (e.g., boron in the chemical shim of PWRs)

fCR: control rod insertion fraction (the fraction of control rod insertion depth:

0 � fCR � 1)

These parameters are given as operation conditions, initial guess values, or

iterative calculation values, and then the macroscopic cross section for the core

calculation is prepared as follows.

• E and R: Use the corresponding macroscopic cross section table.

• E and ρ : Interpolate the macroscopic cross section table in two dimensions.

• ρ, Tf, and Tm: Use the function fitting Eqs. (2.34), (2.35), and (2.36) based onthe branch-off calculation.

• Ni and SB: Correct the macroscopic cross section in the following equations,

using the changes from the condition, in which the homogenized atomic

number density was prepared, and the homogenizedmicroscopic cross section:

ð2:95Þ


ð2:96Þ

where Σ0,x,g ~r; tð Þ is the macroscopic cross section before the correction and

σ ix,g ~r; tð Þ is the homogenized microscopic cross section prepared in the same

way as Eq. (2.92).N i0 E; ρð Þ is the atomic number density homogenized in the

lattice burnup calculation and Ni ~r; tð Þ is the homogenized atomic number

density in the core calculation. For example, the atomic number density of135Xe after a long shutdown is zero and it reaches an equilibrium concen-

tration depending on the neutron flux level after startup. The concentration

of soluble boron is similarly corrected changing the homogenized atomic

number density of 10B

• fCR: Weight and average the macroscopic cross sections ∑ Inx;g and ∑ Out

x;g at

control rod insertion and withdrawal respectively with the control rod

insertion fraction.

ð2:97Þ

Since the cross section prepared in the iteration of the N-TH coupled core

calculation reflects the feedback of instantaneous parameters, it is hereafter

referred to as the feedback cross section. The nuclear calculation is performed

with the feedback cross section by the nodal diffusion method or the finite

difference method and it provides the effective multiplication factor, neutron

flux distribution, power distribution, and so on. The distribution of homoge-

nized atomic number density of nuclides such as 135Xe is calculated from

necessity. For example, since 135Xe has an equilibrium concentration in a

short time after startup as below, it is provided as a homogenized atomic

number density used for the correction of microscopic cross section:

ð2:98Þwhere γXe is the cumulative fission yield of 135Xe and λXe is the decay constant

of 135Xe.

The thermal-hydraulic calculation is performed using the power distribution

acquired from the nuclear calculation and gives the instantaneous moderator

density ρ and the moderator temperature Tm. A BWR fuel assembly is enclosed

in a channel box and the coolant flow inside the assembly is described as a 1D

flow in a single channel with a hydraulic equivalent diameter (the “single

channel model”). The core is modeled as a bundle of single channels, which

are connected at the inlet and outlet, corresponding to each fuel assembly.

This is referred to as the 1D multi-channel model. The calculation procedure

for ρ and Tm in the 1D multi-channel model is as below.


(i) The total flow rate at the inlet is constant and the inlet flow rate Wi for

channel i is distributed (flow rate distribution). A guessed value is given

to Wi if the flow rate distribution is unknown.

(ii) The axial heat generation distribution q0i(z) (assembly linear power) of

the fuel assembly from the nuclear calculation and the enthalpy at inlet

hINi are used to calculate the enthalpy rise in each channel.

ð2:99Þ

(iii) The physical property values at an arbitrary position ~r i; zð Þ such as the

fluid temperature Tm ~rð Þ and ρl ~rð Þ are obtained from the steam table.

(iv) The void fraction distribution α ~rð Þ is acquired using q0i(z), hi(z), physical

property values, and some correlations based on the subcooled boiling

model. The distribution of the instantaneous moderator density ρ ~rð Þ is

calculated using the void fraction distribution.

ð2:100Þ

(v) The pressure drop ΔPi by the channel is calculated using the information

from (i) to (iv) including α ~rð Þ and correlations on pressure drop

(vi) The inlet flow rate is determined so that the pressure drop in all channels

is equal. A new flow rate distribution Wnewi (z) is decided from the flow

rate distribution Wi in (i) and the pressure drop ΔPi in (v).

(vii) The calculations from (i) to (vi) are repeated until the flow rate distribution

is in agreement with the previously calculated one and the pressure drop of

each channel is coincident (iterative calculation for flow rate adjustment).

ρ ~rð Þ and Tm ~rð Þ at a convergence of the iterative calculation are taken and

then the next fuel temperature calculation is done.

In a case such as the space-dependent kinetics calculation, the heat convection

and conduction calculation is done using the same iteration approach of the

N-TH coupled core calculation and then the temperature distribution in the fuel

rod is calculated. In a case such as the steady-state calculation in which operation

conditions are limited, it is common to calculate the fuel temperatures at

representative linear power and fuel burnup in advance and then interpolate the

linear power and burnup obtained from the core calculation into the fuel tem-

perature table to simplify calculation of the fuel temperature Tf ~rð Þ.The calculations mentioned above, the feedback cross section calculation,

the nuclear calculation, the thermal-hydraulic calculation, and the fuel temper-

ature calculation, are repeatedly performed until the effective multiplication

factor and power distribution reach convergence.

[9] Core burnup calculation

Since the time variation of nuclear characteristics in the core is relatively slow

with burnup, the normal N-TH coupled core calculation is carried out using the

time-independent equation at each time step as shown in Fig. 2.26.


~H t0ð Þ represents the distribution of historical parameters in the core at time t0.The burnup E ~r; t0ð Þ and the historical moderator density ρ ~r; t0ð Þ are the typicalhistorical parameters, and other ones can be used depending on reactor design

code. The burnup in conventional LWRs is usually expressed in the unit of

(MWd/t), which is measured as the energy (MWd) produced per metric ton of

heavy metal initially contained in the fuel. That is, the burnup is the time-

integrated quantity of the thermal power.

For the historical moderator density, a spatial distribution of burnup E ~r; t0ð Þat time t0 and a spatial distribution of thermal power density q 000 ~r; t0ð Þ obtainedfrom the N-TH coupled core calculation are considered next. The thermal

power density is assumed not to change much through the interval to the next

time step (t0 � t � t0+ Δt). Then, the burnup distribution at time tN (¼t0 + Δt)can be given by

ð2:101Þ

where C is a constant for unit conversion.

Next, the distribution of the historical moderator density is recalculated.

Since it is the burnup-weighted average value of the instantaneous moderator

density, the historical moderator density at time tN is given by

ð2:102Þ

where E0 and EN are abbreviations ofE ~r; t0ð Þ andE ~r; tNð Þ, respectively. Here, ifthe distribution of instantaneous moderator density obtained from the N-TH

coupled core calculation at time t0 remains almost constant during the time

interval (t0 � t � tN),~ρ ~r; tNð Þ can be expressed as Eq. (2.103).

Time (t)

t0 tN = t0 + Δt

H(t0)

)(t0IH (tN)

H

I

: Historical Parameters

: Instantaneous Parameters

Fig. 2.26 Renewal of historical parameters in core burnup calculation


ð2:103Þ

Hence, the historical moderator density at the next time step tN can be

calculated using the historical moderator density and the instantaneous mod-

erator density obtained from the N-TH coupled core calculation, at time t0.Thus, the core burnup calculation can proceed until the target burnup by

renewing the historical parameters with the burnup step.

[10] Space-dependent kinetics calculation

A space-dependent transient analysis for a short time is made using the time-

dependent diffusion equation as

ð2:104Þ

where β is the delayed neutron fraction, χpg is the prompt neutron spectrum,

and χdi;g is the neutron spectrum of delayed neutron group i. Ci is the precursor

concentration of delayed neutron group i and λi is its decay constant. Com-

parison with the normal multi-group diffusion equation [see Eq. (2.90)] shows

that the time derivative term in the LHS is added and a different expression of

the fission source is given by the fourth and fifth terms of the RHS. Both terms

represent the prompt and delayed neutron sources which are classified by time

behavior. The precursor concentration balance equation can be given by

ð2:105Þ

For an extremely short time difference of 10�4 to 10�3 (t ¼ told + Δt), thefollowing approximations are introduced.

ð2:106Þ

ð2:107Þ


The macroscopic cross section in Eq. (2.104) is the feedback cross section

similar to that in the N-TH coupled core calculation. Since Δt is very short, themacroscopic cross section at time t can be substituted by the macroscopic

cross section corresponding to the instantaneous parameters at t ¼ told.

ð2:108Þ

Substitution of Eqs. (2.106) and (2.107) into Eqs. (2.104) and (2.105) gives

Eq. (2.109).2

ð2:109Þ

This is in the same form as that of the steady-state multi-group diffusion

equation for a fixed-source problem and it can be solved for ϕg ~r; tð Þ by the

numerical solutions mentioned until now. Then, the thermal-hydraulic calcu-

lation is performed and the instantaneous parameters are recalculated, simi-

larly to the steady-state N-TH coupled calculation. The feedback cross section

at the next time step is in turn prepared from Eq. (2.108) and the diffusion

equation of Eq. (2.109) is solved again. These repeated calculations give the

neutron flux distribution and its corresponding thermal power distribution at

each time step.

Since it is hard to use a fine time interval less than 10�3 s in a practical code

for the space-dependent kinetics calculation, a large number of considerations

have been introduced for high-speed and high-accuracy calculation within a

practical time [19]; two examples are the method to express the time variation

in neutron flux as an exponential function and the method to describe the

neutron flux distribution as the product of the amplitude component with fast

time-variation and the space component with relatively smooth variation (the

improved quasi-static method). The fast nodal diffusion method is a general

solution in the analysis code for LWRs.

The point kinetics model widely used is in principle based on the first order

perturbation theory where it is assumed that the spatial distribution of neutron

flux does not change much even though the cross section varies by some

external perturbation. Hence, when the neutron flux distribution is consider-

ably distorted due to such an event as a control rod ejection accident, the

space-dependent kinetics analysis must accurately predict the time behavior of

the reactor.

2 For simplicity, it is assumed that χdi;g ¼ χg which is reasonable for the two-group problem.


2.2 Reactor Core, Plant Dynamics and Safety Calculations

Nuclear reactor design and analysis is done with knowledge and data from funda-

mental fields such as nuclear reactor physics, nuclear reactor kinetics and plant

control, nuclear thermal-hydraulic engineering, nuclear mechanical engineering,

and nuclear safety engineering. There are various calculation models from simple

to detailed, and their selection depends on the purpose. Simple models have the

advantage of giving a better understanding of calculation results and physical

phenomena, and easy improvement of calculation codes. With today’s high com-

putational performance, it is common to use a calculation code based on detailed

models from the viewpoint of generic programming. Since the accuracy and

application range of calculation results depend on models and data, it is necessary

to imagine the physical phenomena and comprehend the validity of the calculation

results based on the fundamental knowledge concerning the calculation target.

The nuclear reactor calculation is classified broadly into the reactor core calcula-

tion and the nuclear plant characteristics calculation. The former is done to clarify

nuclear, thermal, or their composite properties. The latter is done to clarify dynamic

and control properties, startup and stability, and safety by modeling pipes and valves

of the coolant system, coolant pump, their control system, steam turbine and con-

denser, etc. connected with the reactor pressure vessel as well as the reactor core.

2.2.1 Reactor Core Calculation

The reactor core calculation is carried out by the N-TH coupled core calculation,

presented in the list [8] of Sect. 2.1.5, to evaluate properties in normal operation of

the reactor. The concept of core design calculation and the calculation model are

discussed here.

[1] Heat transfer calculation in single channel model

Figure 2.27 depicts a 1D cylindrical model to describe one fuel rod and its

surrounding coolant with an equivalent flow path. It is called the single channelmodel and it is the basic model used in the core thermal-hydraulic calculation

and the plant characteristics calculation. The thermal-hydraulic properties in

the core can be calculated on the single channel model where the heat generated

from fuel pellets is transferred to coolant which is transported with a temper-

ature rise.

The radial heat transfer model is composed of fuel pellet, gap, cladding, and

coolant. The radial heat conduction or convection between those components is

considered in each axial region and then the axial heat transport by coolant or

moderator is examined. The mass and energy conservation equations and the

state equation are solved in each axial region in turn from the top of the upward

coolant flow. The momentum conservation equation is also solved to evaluate

the pressure drop. The axial heat conduction of the fuel pellet is ignored. This


assumption is valid because the radial temperature gradient is several orders of

magnitude larger than the axial one.

The difference between the fuel average temperature and the cladding

surface temperature is expressed as

ð2:110Þ

where,

kf: average thermal conductivity of pellet (W/m-K)

hg: gap conductance (W/m2-K)

kc: thermal conductivity of cladding (W/m-K)

Q00: heat flux from pellet (W/m2)

rf : pellet radius (m)

tc: cladding thickness (m)

Tavef : pellet average temperature (K)

Ts: cladding surface temperature (K).

The terms inside the RHS square brackets represent the temperature drop in

fuel, gap, and cladding, respectively.

The heat transfer from cladding surface to coolant is described by Newton’slaw of cooling

ð2:111Þ

where,

hc: heat transfer coefficient between cladding surface and coolant (W/m2-K)

rc: cladding radius (m)

Fig. 2.27 Single-channel heat transfer calculation model


Ts: cladding surface temperature (K)

T: coolant bulk temperature (K).

The heat transfer coefficient hc can be evaluated using the Dittus-Boeltercorrelation for single-phase flow and the Thom correlation or the Jens-Lottescorrelation for nucleate boiling.

Water rods are often implemented into fuel assemblies, especially for BWRs.

The heat transfer characteristics of the water rod can also be calculated using

the single channel model. Figure 2.28 describes the heat transfer calculation

model with two single channels of fuel rod and water rod. The heat from

coolant is transferred through the water rod wall into the moderator.

The heat transfer of the water rod is also represented by Newton’s law of

cooling. The temperature difference between the coolant in the fuel rod

channel and the moderator in the water rod channel is expressed as

ð2:112Þ

where,

T: coolant temperature in fuel rod channel (K)

Tw: moderator temperature in water rod channel (K)

Dw: hydraulic diameter of water rod

hs1: heat transfer coefficient between coolant and outer surface of water rod

(W/m2-K)

hs2: heat transfer coefficient between inner surface of water rod and moderator

(W/m2-K)

Nf: number of fuel rods per fuel assembly

Nw: number of water rods per fuel assembly

Q ’ w: linear heat from coolant to water rod (W/m)

tws: thickness of water rod (m)

Fig. 2.28 Single-channel thermal-hydraulic calculation model


The terms inside of the RHS square brackets represent the heat transfer from

the fuel rod channel to the water rod wall and back to the water rod, respec-

tively. Temperature drop due to the water rod wall is ignored in this equation. It

is also assumed that the water rod wall is an unheated wall, whereas the

cladding is a heated wall. Hence, there is no boiling at the outer surface of

the water rod although the outer surface of the water rod wall contacts with two

phase coolant. It should be subsequently noted that the application condition of

heat transfer correlations is not identical for such a water rod wall.

The thermal conductivity of the fuel pellet depends on temperature and in a

BWR fuel, for example, it can be given by

ð2:113Þ

where Tavef and kf are the average temperature (K) and thermal conductivity

(W/m-K) of pellet, respectively. The thermal conductivity is actually a function

of pellet density, plutonium containing fraction, burnup, etc. as well as tem-

perature. Since the fission gas release during irradiation causes pellet swelling

and hence it also changes the thermal gap conductance with cladding, the fuel

behavior analysis code which includes irradiation experiments is required to

precisely evaluate the fuel centerline temperature. However, since the fuel

centerline temperature at normal operation is far lower than the fuel melting

point, it does not need a highly accurately estimate in the reactor core design. In

BWR fuel, the fuel centerline temperature at normal operation is limited to

about 1,900 �C to prevent excessive fission gas release rate. The temperature

drop in the pellet depends not on pellet radius, but linear power density and fuel

thermal conductivity. Hence, the linear power density is restricted at normal

operation of LWRs.

[2] 3D nuclear and thermal-hydraulic coupled core calculation

In nuclear reactors such as LWRs, in which the coolant serves as moderator, the

moderator density varies with temperature change and boiling of coolant, and

subsequently the reactivity and reactor power vary. The power distribution also

varies with the moderator density change. The heat transfer calculation is

performed to evaluate those core characteristics using the heat generation

distribution in the core acquired from the nuclear calculation. The nuclear

calculation is performed again using the macroscopic cross section changed

by the coolant or moderator density distribution obtained in the heat transfer

calculation. These calculations are repeated until a convergence. This is the

nuclear and thermal-hydraulic coupled core calculation (the N-TH coupled core

calculation) mentioned earlier.

A 2D core model cannot precisely handle power and burnup in each part of the

core. This usually needs a 3D nuclear calculation, which is combined with the

heat transfer calculation in the single channel model for each fuel assembly.

Figure 2.29 shows a 3D N-TH coupled core calculation model which is a

symmetric quadrant. Since the fuel assembly burnup differs with the power


history at each position, the fuel assembly is axially segmented and the macro-

scopic cross section at each section is prepared for the core calculation. Each fuel

assembly, which consists of a large number of fuel rods, is described as the

single-channel thermal-hydraulic calculation model. The heat transfer calculation

is carried out with the axial power distribution of each assembly obtained from

the nuclear calculation and it provides an axial moderator density distribution.

The N-TH coupled core calculation above is repeatedly continued until it

satisfies a convergence criterion and then it gives the macroscopic cross section

which is used for calculations such as the core power distribution. This process

is repeated at each burnup step and is referred to as the 3D nuclear and thermal-

hydraulic coupled core calculation (3D N-TH coupled core calculation). In the

practical calculation, as mentioned in Sect. 2.1, the macroscopic cross sections

are tabulated in advance with respect to parameters such as fuel burnup and

enrichment, moderator density, and so on. Each corresponding cross section is

used in the N-TH coupled core calculation.

Fig. 2.29 3D nuclear and thermal-hydraulic coupled core calculation model


Figure 2.30 shows a flow chart of an equilibrium core design using the 3D

N-TH coupled core calculation [25]. Macroscopic cross section data are previ-

ously prepared from the lattice and/or assembly burnup calculation and then

tabulated with fuel burnup and enrichment, moderator density, and so on. The

macroscopic cross section is used in the 3D multi-group diffusion calculation to

obtain the core power distribution, which is employed again in the thermal-

hydraulic calculation.

Usually several different types of fuel assemblies are loaded into the core

according to the purpose such as radial power distribution flattening or neutron

leakage reduction. In the first operation of a reactor, all fresh fuel assemblies are

loaded but they may have several different enrichments of fuel. This fresh core

is called the initial loading core. After one cycle operation, low reactivity

assemblies representing about 1/4 to 1/3 of the loaded assemblies are

discharged from the reactor. The other assemblies are properly rearranged in

the core considering the burnup distribution and other new fresh ones are

loaded.

In the end of each cycle, the core is refueled as in the above way and then

the reactor restarts in the next cycle. While such a cycle is being repeated, the

characteristics of the core vary but gradually approach equilibrium quantities.

When finally the characteristics at N + 1 the cycle have little change in

comparison with those at the N cycle, the core is called the equilibriumcore. In certain cases, it shows equilibrium characteristics at N and N + 2

cycles and these are also generally referred to as the equilibrium core. By

contrast, the core at each cycle is called the “transition core” until it reaches

the equilibrium core.

Fig. 2.30 Flow chart of equilibrium code design


In the equilibrium core design, first a proper burnup distribution of the core is

guessed at the beginning of the cycle (BOC), and a control rod and fuel loading

pattern is assumed. Then, the N-TH coupled core calculation is performed for

one cycle based on the assumed core design with the burnup distribution. After

the core calculation, the refueling according to the fuel loading pattern is

followed by evaluation of the new burnup distribution at BOC of the next

cycle. Such evaluation after one cycle calculation is repeated until the burnup

distribution at BOC converges, at which time it is regarded as attaining the

equilibrium core.

The core design criteria, such as shutdown margin, moderator density coef-

ficient, Doppler coefficient, fuel cladding temperature, and maximum linear

power density, are applied to the equilibrium core. If the equilibrium core does

not satisfy the core design criteria, the equilibrium core design is performed

again after reviewing fuel enrichment zoning, concentration and number of

burnable poison rods, fuel loading and control rod pattern, coolant flow rate

distribution, and so on.

A 3D N-TH coupled core calculation is also performed for fuel management

such as fuel loading and reloading. In this case, the equilibrium core is searched

from an initial loading core. Some fuel assemblies are replaced with new fresh

ones after the first cycle calculation and the new core configuration is consid-

ered in the next cycle calculation. The calculation procedure is also similar to

Fig. 2.30.

It is noted again that operation management such as fuel loading pattern and

reloading, and control rod insertion can be considered in the 3D N-TH coupled

core calculation. Figure 2.31 shows an example of 3D core power distribution

and coolant outlet temperature distribution.

In the core design and operation management, it is necessary to confirm that

maximum linear power density (or maximum linear heat generation rate,

MLHGR) is less than a limiting value. Figure 2.32 shows variation in

MLHGR and maximum cladding surface temperature (MCST) with

cycle burnup, which was from a 3D N-TH coupled core calculation for a

supercritical water-cooled reactor (SCWR) as an example [26]. The limiting

values at normal operation are 39 kW/m and 650 �C respectively in this

example. The core calculation was performed according to the fuel loading

and reloading pattern as described in Fig. 2.33. It is noted that the third cycle

fuel assemblies are loaded in the core peripheral region to reduce neutron

leakage (low leakage loading pattern). The control rod pattern is shown in

Fig. 2.34.

The 3D N-TH coupled core calculation has been done for BWRs for many

years. Since there are not many vapor bubbles in PWRs due to subcooled

boiling, the N-TH coupled core calculation has not always been necessary,

but it is being employed recently for accuracy improvement.


Fig. 2.33 Example of fuel loading and reloading pattern

Fig. 2.32 Example of Variation in maximum linear power density and maximum cladding surface

temperature with cycle burnup

Fig. 2.31 3D core power distribution and coolant outlet temperature distribution

In the core calculation, one fuel assembly is modeled not as a bundle of its

constituent fuel rods but a homogenized assembly, which is represented as a

single channel in the thermal-hydraulic calculation. However, the power of

each fuel rod is actually a little different and hence it leads to a different coolant

flow rate and a different coolant enthalpy in the flow path (subchannel) between

fuel rods. This also causes a difference such as in cladding temperature which is

particular to a single-phase flow cooled reactor. The heat transfer analysis for

coolant flow paths formed by many fuel rods is called subchannel analysis, andit is used in evaluation of cladding temperature in liquid sodium-cooled fast

reactors and so on.

In computational fluid dynamics (CFD), the behavior of fluid is evaluated

through numerical analysis of the simultaneous equations: Navier–Stokes

equations, continuity equation, and in some cases perhaps the energy conser-

vation equation and state equation. It subsequently requires a huge computing

time, but a detailed heat transfer analysis can be made. The CFD analysis can be

used to calculate a circumferential distribution of fuel rod temperature in a

single-phase flow cooled reactor or to analyze detailed fluid behavior inside the

reactor vessel and pipes. Figure 2.35 depicts the three heat transfer analysis

models mentioned above including the CFD analysis model.

Fig. 2.34 Example of control rod pattern and insertion


2.2.2 Plant Dynamics Calculation

[1] Plant dynamics calculation code

The plant dynamics calculation treats plant control, stability, and response at

abnormal transients and accidents. A simple model for heat transfer calculation

in a plant is the node junction model as shown in Fig. 2.36 [27]. In the node

junction model, reactor components are represented as 1D nodes and

connected; the connected items include core, upper and lower plena,

downcomer (inlet coolant flows down the downcomer region which is placed

between the core and reactor pressure vessel), main feedwater and main steam

lines, valves, etc. The node junction calculation begins from an upstream node

based on mass and energy conservation laws. The momentum conservation law

is also considered for pressure drop and flow rate distribution calculations. In

the case of Fig. 2.36, for example, the core is described with two single channel

models at average and maximum power. The core design calculation is

performed at steady state as mentioned before, but a transient single channel

model is used in the plant dynamics calculation.

A transient radial heat conduction in fuel can be expressed as

ð2:114Þ

Fig. 2.35 Heat transfer analysis models


where,

Cp: specific heat of pellet (J/kg-K)

kf: thermal conductivity of pellet (W/m-K)

qm: power density (W/m3)

r: radial distance (m)

Tf: pellet temperature (K)

ρf: pellet density (kg/m3).

The heat transfer from cladding outer surface to coolant is evaluated by

Newton’s law of cooling. A proper heat transfer coefficient should be used

considering flow regime and temperature, pressure, etc.

Plant dynamics analysis codes are constructed with the node junction model

and the following reactor kinetics model. The simplest kinetics model is the

point reactor kinetics model (point reactor approximation) as

ð2:115Þ

Fig. 2.36 Node junction model


where,

n(t): number of neutrons

Ci(t): number of delayed neutron precursor for group it: time

βi: fraction of delayed neutron group i

β:

Δρ: reactivityΛ: prompt neutron generation time (s)

λi: decay constant of precursor of delayed neutron group i (s�1)

Tavef (t): average fuel temperature (K)

ρmod(t): moderator density (g/cm3).

The reactivity feedback of fuel temperature (Doppler effect) and moderator

density is applied to the point reactor kinetics equations. Even though large

power reactors such as LWRs show space-dependent kinetic characteristics, the

reactivity feedback effect can be expressed with the point reactor approxima-

tion using a space-dependent weight function (adjoint neutron flux).

Figure 2.37 shows an example of a node junction model in a plant dynamics

analysis code and its calculation flow chart. This model describes coolant flow

which is fed to the water moderation rods through the top dome of the reactor

vessel for a supercritical water-cooled thermal reactor (Super LWR) [28, 29].

The mass, energy, and momentum conservation equations used in the

node junction model are a time-dependent form of the single-channel

Fig. 2.37 Plant dynamics analysis model and calculation flow chart


thermal-hydraulic model in the core design. In the case of single channel, single

phase, and one dimension, the mass conservation equation is given by Eq. (2.116)

ð2:116Þ

and the energy conservation equation is given by Eq. (2.117).

ð2:117Þ

The momentum conservation equation is expressed as

ð2:118Þ

and the state equation is

ð2:119Þwhere,

t: time (s)

z: position (m)

ρ: coolant density (kg/m3)

u: fluid velocity (m/s)

h: specific enthalpy (J/kg)

q00: heat flux at fuel rod surface (W/m2)

A: flow area of fuel channel (m2)

Pe: wetted perimeter of fuel rod (m)

P: pressure (Pa)g: gravitational accelerationDh: hydraulic equivalent diameter of fuel channel (m)

Re: Reynolds number

θ: vertical angle of fuel channelf: frictional coefficientfor example, f ¼ 0.0791 � Re�0.25 (Blasius equation).

In the case of a water rod, the energy transferred to the water rod should be

considered in the energy conservation equation of the fuel rod cooling channel.

In the case of light water, the steam tables of the Japan Society of Mechanical

Engineers (JSME) or the American Society of Mechanical Engineers (ASME)

can be used in the state equation of the coolant.


The following introduce heat transfer coefficients for water-cooled reactors.

In a single-phase turbulent flow, for example, the heat transfer coefficient can

be obtained from the Dittus-Boelter correlation

ð2:120Þ

where,

De: hydraulic equivalent diameter (m)

G: mass flux (kg/m2-s)

h: heat transfer coefficient (W/m2-K)

k: thermal conductivity (W/m-K)

Pr: Prandtl number

μ: viscosity coefficient (Pa-s).

If nucleate boiling occurs, the heat transfer for PWR can be described by the

Thom correlation

ð2:121Þ

where,


P: pressure (MPa)

dT: temperature difference between wall surface and fluid (K)

ΔTsat: wall temperature elevation above saturation temperature (K)

and it is applied for the range of

pressure: 5.2–14.0 MPa

mass flux: 1,000–3,800 kg/m2-s

heat flux: 0–1600 kW/m2.

The Jens-Lottes correlation can be used for BWRs.

If there is a thin liquid film in annular flow with a high steam quality, for

example, the heat transfer to a vertical upward water flow can be represented by

the Schrock-Grossman correlation

ð2:122Þ

where,





kf: thermal conductivity of water (W/m-K)

Prf: Prandtl number of water

X: steam quality

ρf: water density (kg/m3)

ρg: steam density (kg/m3)

μf: viscosity coefficient of water (Pa-s)

μg: viscosity coefficient of steam (Pa-s)

and it is applied for the range of

pressure: 0.3–3.5 MPa

mass flux: 240–4,500 kg/m2-s

heat flux: 190–4,600 kW/m2.

In the post-dryout region of BWRs, namely, the steady-state film boiling

state of mist flow where steam flow is accompanied with liquid droplets, for

example, the heat transfer can be described by the Groeneveld correlation.

ð2:123Þ

where,


kg: thermal conductivity of saturated steam (W/m-K)

Prv,w: Prandtl number of steam at wall surface



μg: viscosity coefficient of steam (Pa-s)

X: steam quality

ρf: water density (kg/m3)

ρg: steam density (kg/m3)

[2] Plant control

The plant dynamics analysis covers the plant control system. Figure 2.38 pre-

sents the plant control system of a SCWR as an example [27]. Typical control

systems of LWRs are actually more complex, but they are based on the concept

below. The plant control system of nuclear power plants uses proportional-integral-derivative (PID) control which is widely used in the control system of

other types of practical plants as well.

Composition of the plant control system is based on similar existing plant

control systems and its parameters are determined using plant dynamics anal-

ysis codes using the procedure below.


① Investigate responses to external disturbances without control system, and

decide what state variables (e.g., power) to control and how to control them

(e.g., by control rod) according to their sensitivities.

② Design a control system.

③ Confirm responses to the external disturbances under the control system.

For example, in the case of reducing feedwater flow rate, changing control

rod position, inserting a reactivity, or closing main steam control valves,

variations in main steam temperature, power, or pressure are investigated for

each case using plant dynamics analysis codes. In Fig. 2.38, the main steam

temperature is controlled by regulating feedwater flow rate, the reactor power

by control rods, and the main steam pressure by main steam control valves

considering highly sensitive parameters.

Figure 2.39 shows the power control system by control rods. This control

system is governed by a proportional controller which decides a control rod

drive speed in proportion to the deviation between actual power and power

setpoint. Control rods are driven at a maximum speed for a deviation larger than

a constant value b.Figure 2.40 represents the main steam temperature control system which is

driven by a proportional-integral (PI) controller after a lead compensation for

time-lag of a temperature sensor. The PI control parameter values are deter-

mined for stable and fast-convergence responses through fine tuning of the

integral gain with a little change of the proportional gain.

Fig. 2.38 Plant control system (supercritical water-cooled reactor)


Figure 2.41 shows the pressure control system which controls valve opening

by converting the deviation from the pressure setpoint into a valve opening

signal with lead/lag compensation. The control parameter value is determined

for a stable convergence response from investigation of a time variation in the

main steam pressure with a little change of the pressure gain.

Fig. 2.39 Power control system

Fig. 2.40 Main steam temperature control system


After all control systems are set up, the plant behavior with external distur-

bances of power, pressure, and flow rate is analyzed using the plant dynamics

analysis code until finally stable and fast convergence is confirmed.

[3] Plant startup

Since the plant startup and stability show a slow behavior change over a long

time, the balance of plant (BOP) such as the turbine system is included in the

calculation model of plant dynamics calculation codes. Figure 2.42 shows an

example of the calculation model for a SCWR [28]. The BOP can be modeled

as properly describing mass and energy conservation and plant time behavior

without a detailed embodiment. The reactor core is represented as a single

channel model in the figure, but it can be described by two single channels,

i.e. average and hot channels.

The following are models for pressure change in inlet orifice, feedwater

pump, feedwater pipe, and outlet valve used in the ex-core circulation system.

Fig. 2.41 Pressure control system


P: pressure (Pa)ΔP: pressure drop (Pa)

δP: pressure change in pump (Pa)

ζ: orifice or valve pressure drop coefficient

ρ: coolant density (kg/cm3)

u: fluid velocity (m/s)

δu: fluid velocity change in pump

Cpump: pressure drop coefficient of pump

z: position (m)

t: time (s)

f: friction pressure drop coefficient

D: diameter of feedwater pipe (m)

The plant startup procedure is established to satisfy constraints at startup

using the calculation models above. The constraints at startup depend on reactor

type: for example, maximum cladding surface temperature for SCWRs; restric-

tions in heat flux, various safety parameters, and pump cavitation for LWRs;

restrictions of vapor fraction in main steam for direct-cycle reactors. There are

also restrictions from thermal stress to temperature rise of the reactor vessel.

Fig. 2.42 Calculation model for plant startup and stability analysis


[4] Reactor stability analysis

In the reactor stability analysis, first governing equations are established for

normal and perturbed values of state variables from the plant dynamics codes

and then the governing equations are linearized for the perturbed one. The

equations are converted into a frequency domain by the Laplace transform and

analyzed there. This is referred to as linear stability analysis and the procedurefor it in the frequency domain is shown in Fig. 2.43.

In the figure, the system transfer function (closed-loop transfer function) is

defined with the open-loop transfer functions, G(s) and H(s) The system

stability is characterized by the poles of the transfer function (the roots of the

characteristic equation in its denominator) and described by the decay ratio.

The concept and the stability criteria are given in Fig. 2.44. For the system to be

stable with damping, all the roots of the characteristic equation must have

negative real parts. The decay ratio, which is defined as the ratio of two

consecutive peaks of the impulse response of the oscillation for the represen-

tative root (the root nearest to the pole axis), depends on the calculation mesh

size. The decay ratio is determined by extrapolation to zero mesh size following

calculations with different mesh size.

The following should be considered in the stability analysis of nuclear

reactor design.

(i) Channel stability: This is thermal-hydraulic stability of the fuel cooling

channel.

(ii) Core stability: This is the nuclear and thermal-hydraulic coupled stability

where the whole core power (neutron flux) regularly oscillates as a

fundamental mode due to the moderator density feedback

Fig. 2.43 Procedure of linear stability analysis


(iii) Regional stability: This is a kind of core stability and the neutron flux of

each region oscillates as a high-order mode by reciprocally going up

and down.

(iv) Plant stability

(v) Xenon stability (Xe spatial oscillation): This is a function of accumulation

and destruction of FP Xe, and spatial change of neutron flux

The channel stability is thermal-hydraulic stability of single coolant channel

in the core. The core stability is the nuclear and thermal-hydraulic coupled

stability where the whole core power (neutron flux) regularly oscillates as a

fundamental mode due to the moderator density feedback. The regional stabil-

ity is a kind of core stability and the neutron flux of each region oscillates as a

high-order mode by reciprocally going up and down. These three stabilities are

inherent characteristics of nuclear reactors with a large change in moderator

density in the core such as BWRs, and hence they are evaluated by the

frequency-domain stability analysis. The plant stability is the stability of

plant system including its control system and is evaluated by time-domain

analysis using a plant dynamics analysis code. Xenon spatial oscillation is

caused by accumulation and destruction of Xenon-135, and spatial change of

neutron flux. The xenon stability is analyzed by the production and destruction

equation of Xenon-135.

Fig. 2.44 Decay ratio and stability criteria


2.2.3 Safety Analysis

[1] Analysis of abnormal transients and design based accidents

In the safety analysis of nuclear power plants, safety system models and

scenarios of abnormalities are implemented into the plant dynamics calculation

code. Abnormal events are classified into abnormal transients and design basis

accidents depending on the initial event frequency, namely, high and low

frequencies respectively.

For examples, Fig. 2.45 describes the plant system of a SCWR with the

safety system and Fig. 2.46 shows the calculation model of its safety analysis

code [27, 29].

The SCWR is a direct steam cycle system and its safety system is similar to

that of BWRs. Safety relief valves (SRVs) and an automatic depressurization

system (ADS) are installed into the main steam lines. Turbine bypass valves are

used to dump the main steam to the condenser at a sudden closure of the main

steam control valves. This is made in a manner to prevent damage to the turbine

from over-rotation by a turbine load loss due to a breakdown of the electric

transmission system. The high-pressure auxiliary feedwater system (AFS, high-

pressure coolant injection system) is equipped to maintain coolant supply at an

abnormal event of the main feedwater system. The low pressure coolant

injection (LPCI) system is prepared to flood and cool the core after reactor

depressurization at a loss of coolant accident (LOCA). The water is fed from the

suppression chamber or condensate water storage tank. The LPCI system also

Fig. 2.45 Plant and safety system (SCWR)


has the function of residual heat removal (RHR) after reactor shutdown. The

boric acid injection system (standby liquid control system, SLC system) is

provided for backup shutdown.

Figure 2.46 models main steam control valves (turbine control valves), SRVs,

turbine bypass valves, AFS, reactor coolant pump (RCP), and so on. The core is

modeled by average and hot channels. Since abnormal events for safety analysis

are often relatively short-term ones, heat removal models including the turbine

system are not always necessary. The calculation model in the figure does not

comprise safety systems for LOCA analysis such as the LPCI. General-purpose

safety analysis codes have models to handle those systems.

[2] Loss of coolant accident

LOCA analysis is composed of two parts: blowdown phase and reflood phase.

As an example of SCWR, the blowdown analysis model is shown in Fig. 2.47 in

which valves and systems able to simulate cold-leg break and hot-leg break are

implemented into the abnormal transient and accident analysis model of Fig. 2.46.

Fig. 2.46 Calculation model for abnormal transient and accident analysis


The reflood phase can be analyzed by modeling the LPCI in the calculation

code as shown in Fig. 2.48. In the practical analysis of plant safety, general-

purpose codes are used to describe more detailed models and to calculate the

blowdown and reflood phases as one body.

2.2.4 Fuel Rod Analysis

[1] Fuel rod integrity

The principle role of fuel rod cladding is to confine radioactive FPs and to

prevent contamination of the coolant system; therefore an assurance of fuel rod

integrity is important for reactor design and safety.

When an abnormal transient event occurs in a nuclear plant with a frequency

of more than once during the plant life time, the safety criterion is associated

with whether the core can return to the normal operation without core damage

Fig. 2.47 Calculation model for LOCA blowdown analysis


after the plant is stabilized. Thus, fuel rods are required to keep their integrity

during abnormal transients (anticipated operational occurrences) as well as

normal operation.

The criteria for fuel rod integrity differ somewhat between BWRs and PWRs

because of different operating conditions. Fuel rods are designed generally

based on the following criteria [25, 27].

(i) Average circumferential plastic deformation of fuel cladding < 1 %

(ii) Fuel centerline temperature < Pellet melting point (PWR)

(iii) No overpressure within fuel rod

(iv) Allowable stress of fuel rod cladding

(v) Cumulative damage fraction < 1

The overall fuel rod integrity is evaluated further in cladding corrosion and

hydriding, pellet cladding interaction (PCI), cladding creep rupture, rod fretting

wear, cladding bending, and irradiation growth of the fuel rod and assembly.

The fuel damage criteria and allowable limits of BWRs and PWRs are given

in Table 2.2. Based on the system of fuel damage occurrence in BWRs or

PWRs, restrictions at normal operation and allowable limits at abnormal tran-

sient are determined in the fuel design considering the each following;

(i) Cladding damage due to overheating resulting from insufficient cooling

(ii) Cladding damage due to deformation resulting from a relative expansion

between pellet and cladding

Fig. 2.48 Calculation model for LOCA reflooding analysis


For damage mechanism (i), the BWR criterion is that “the fuel cladding

integrity ensures that during normal operation and abnormal transients, at least

99.9 % of the fuel rods in the core do not experience transition boiling”.

In PWRs, similarly, the criterion is that “the DNB design basis requires at

least a 95 % probability, at a 95 % confidence level, that the limiting fuel rods in

the core will not experience DNB during normal operation or any transient

conditions”. (DNB means the departure from nucleate boiling.)

Based on those criteria, the BWR design requires the allowable limit of the

minimum critical power ratio (MCPR) to be 1.07 at abnormal transients and

the restriction to be 1.2–1.3 at normal operation. Similarly, the PWR design has

the allowable limit of minimum heat flux ratio (minimum departure from

nucleate boiling ratio, MDNBR) to be 1.30 at abnormal transients and the

restriction to be 1.72 at normal operation.

For the damage mechanism (ii), the BWR criterion is that “During normal

operation and abnormal transients, the fuel rods in the core do not experience

circumferential plastic deformation over 1 % by PCI”. In PWRs, similarly, the

criterion is that “During normal operation and abnormal transients, the fuel rods

Table 2.2 Criteria and allowable limits of fuel damage in BWR and PWR

Damage BWR PWR

(i) Cladding

overheating

Criteria The fuel cladding integrity

ensures that during normal

operation and abnormal

transients, at least 99.9 %

of the fuel rods in the core

do not experience transi-

tion boiling

The DNB design basis

requires at least a 95 %

probability, at a 95 %

confidence level, that the

limiting fuel rods in the

core will not experience

DNB during normal

operation or any transient

conditions

Allowable

limits at

transient

MCPR: 1.07 MDNBR: 1.30

Restrictions at

normal

MCPR: 1.2–1.3 MDNBR: 1.72

(ii) Cladding

deformation

Criteria The fuel cladding integrity



transients, the fuel rods in

the core do not experience

circumferential plastic

deformation over 1 % by

PCI

The fuel cladding integrity



transients, the fuel rods in

the core do not experience

circumferential deforma-

tion (elastic, plastic, and

creep) over 1 % by PCI

Allowable

limits at

transient

Direct evaluation of criteria

by experiments

MLHGR: 59.1 kW/m

(2,300 �C fuel centerline

temperature)

Restrictions at

normal

MLHGR: 44 kW/m MLHGR: 43.1 kW/m

(1,870 �C fuel centerline

temperature)


in the core do not experience circumferential deformation (elastic, plastic, and

creep) over 1 % by PCI”.

Central melting of pellets in LWR fuel causes a phase change and a volume

increase, and the fuel cladding may be substantially deformed mainly due to

pellet cladding mechanical interaction (PCMI). The fuel rod design criteria of

PWRs require that the fuel centerline temperature will be lower than the pellet

melting point. Hence, the allowable limit of the fuel centerline temperature is

determined to be 2,300 �C at abnormal transients and its corresponding max-

imum linear power density is 59.1 kW/m. The restrictions at normal operation

are 1,870 �C for the fuel centerline temperature and 43.1 kW/m of the maxi-

mum linear power density.

For internal pressure of fuel rods, the BWR design requires that the cladding

stress due to the internal pressure will be less than the allowable strength limit.

The PWR design restricts the internal pressure of fuel rods to less than the rated

pressure of primary coolant (157 kg/cm2-g) in order to avoid expansion of the

gap between pellet and cladding due to creep deformation of cladding outside at

normal operation. This phenomenon is called “lift-off”. The criteria of “ASME

B&PV Code Sec III” are used as allowable stress limits for LWRs.

[2] Fuel rod behavior calculation

The fuel rod behavior caused by irradiation of fuel pellets and cladding is

complicated for investigation in an analytical method. Fuel rod behavior

calculation codes were developed from various experiments and operational

experiences. FEMAXI-6 [30] as an open code developed in Japan is a general

analysis code for fuel rod behavior at normal operation or abnormal transients

in LWRs. It constructs a model composed of one fuel pellet, cladding, and

internal gas, and then analyzes thermal, mechanical, and chemical behavior and

reciprocal action at normal operation or abnormal transients from overall power

history data. Table 2.3 shows fuel rod behavior analyzed in FEMAXI-6.

The overall structure of FEMAXI-6 consists of two parts, thermal analysis

and mechanical analysis, as shown in Fig. 2.49. The thermal analysis part

evaluates radial and axial temperature distributions considering a change in

gap size between pellet and cladding, FP gas release models, axial gas flow and

Table 2.3 Fuel rod behavior in FEMAXI-6

Thermal behavior (temperature

dependent) Mechanical behavior

Pellet Thermal conduction (radial heat

flux distribution)

Fission gas release (temperature

and burnup dependent)

Thermal expansion, elasticity and plasticity, creep,

cracking, initial relocation, densification,

swelling (solid FPs and gas bubbles), hot-press

Cladding Thermal conduction

Oxidation film growth

Thermal expansion, elasticity and plasticity, creep,

irradiation growth, oxidation film growth

Fuel Rod Gap thermal conduction (mixed

gas, contact, radiation)

Cladding surface heat transfer

Gap gas flow

PCMI (pellet cladding mechanical interaction),

friction, bonding


its feedback to heat transfer in gap, and so on. The mechanical analysis part

applies the finite element method (FEM) to the whole fuel rod and analyzes

mechanical behavior of fuel pellet and cladding as well as PCMI. It also

calculates an initial deformation due to thermal expansion, fuel densification,

swelling, and pellet relocation, and then calculates stress and deformation of

pellet and cladding considering cracking, elasticity/plasticity, and pellet creep.

The thermal and mechanical analysis is iteratively performed to consider the

thermal feedback effect on the fuel rod mechanical behavior. Further, the local

PCMI can be analyzed by the 2D FEM, based on the calculation results from

both analysis parts such as temperature distribution and internal pressure.

Figure 2.50 describes the calculation model of FEMAXI-6. The entire fuel rod

is divided axially into several tens of segments and radially into ring elements.

For example, the figure shows ten axial and radial divisions in the fuel pellet.

Fig. 2.49 Overview of FEMAXI-6 structure


FEMAXI-6 contains various calculation models and experimental correla-

tions that users can specify. Details of models, physical properties, or correla-

tions are described in the code manual [30] of FEMAXI-6.

Exercises of Chapter 2

[1] Consider a PWR UO2 pellet 0.81 cm in diameter, 1.0 cm in height, and

10.4 g/cm3 in density in which the uranium is enriched to 3.2 wt. %235U. Answer the following questions.

(a) Calculate the atomic number densities of 235U, 238U, and O in the pellet.

UO2 contains only 235U, 238U, and O. Their atomic masses are M(235U) ¼ 235.04, M(238U) ¼ 238.05, and M(O) ¼ 16.0, and Avogadro’s

number is 0.6022 � 1024.

(b) One-group microscopic cross sections of 235U, 238U, and O are given in the

following table.

Nuclide Absorption σa (barn) Fission σf (barn)235U 50 41238U 1.00 0.10

O 0.0025 0.00

(1 barn ¼ 10�24 cm2)

Compute the macroscopic absorption and fission cross sections of the

pellet. What is the probability that a neutron will cause a fission on being

absorbed into the pellet?

Fig. 2.50 FEMAXI-6 calculation model


(c) Compute the mass (in tons) of metal uranium in one pellet 1.0 cm in height.

A PWR fuel assembly consists of 265 fuel rods bundled into a 17 � 17

array (including 24 guide tubes) with an active height of 366 cm. Compute

the initial mass of uranium in one fuel assembly and compute the total

loading amount of uranium (the mass of metal uranium) in the core which

contains the 193 fuel assemblies.

(d) Calculate the linear power density (W/cm) of the fuel rod when one-group

neutron flux in the pellet is 3.2 � 1014 n/cm2-s. The energy released per

fission is 200 MeV (1MeV ¼ 1.602 � 10�13 J). What is the specific power

[power per initial mass of metal uranium (MW/ton)]?

(e) Compute the burnup (MWd/ton) of the pellet consumed along the specific

powers as shown in the figure

(f) Compute the consumed amount of 235U in the pellet from the burnup of (e).

Two-thirds of the total power is generated by the uranium (235U and 238U)

fission and the rest by the fission of plutonium produced from the conver-

sion of 238U.

[2] An infinite slab of non-multiplying uniform medium of thickness d (¼ 400 cm)

contains a uniformly distributed neutron source S0 as shown in the following

figure.


The neutron flux distribution in the slab can be determined by solving the

one-group diffusion equation

where D ¼ 1.0 cm, Σa ¼ 2.5�10�4 cm�1, and S0 ¼ 1.0 � 10�3 n/cm2-s.

Discretize one side to the reflective boundary in the middle into four meshes

(N ¼ 4) and calculate the neutron flux distribution using the finite difference

method. Plot the distribution and compare it with the analytical solution

Calculate for N ¼ 8 and 16 if programming is available.

[3] (a) Consider a multiplying system with an effective multiplication factor keff.Develop the diffusion equations for fast and thermal groups in terms of the

two-group constant symbols in the following table.

Fast group Thermal group

χ1 1.00 χ2 0.00

ν1 2.55 ν2 2.44

Σf, 1 2.76 � 10�3 Σf, 2 5.45 � 10�2

Σa, 1 9.89 � 10�3 Σa, 2 8.20 � 10�2

Σ1!2 1.97 � 10�2 Σ2!1 0.00

D1 1.42 D2 0.483

(b) Calculate the infinite multiplication factor k1 using the two-group

constants.

[4] The burnup chain of 135I and 135Xe is given by the following figure

Show that the equilibrium concentration of 135Xe (NXe1 ) in the reactor

operating at a sufficiently high neutron flux is given approximately by


where γI and γXe are the fission yields of 135I and 135Xe respectively, and σXea is

the one-group microscopic absorption cross section of 135Xe and Σf is the

one-group macroscopic fission cross section.

[5] The energy conservation equation is given by Eq. (2.117) for no water moder-

ation rod. Develop the energy conservation equation when introducing the

water moderation rod.

[6] Develop equations for single-phase transient analysis at the 1D axial node i of asingle coolant channel, based on the mass conservation equation [Eq. (2.116)],

the energy conservation equation [Eq. (2.117)], the momentum conservation

equation [Eq. (2.118)], and the state equation [Eq. (2.119)], referring to the

figure.

References

1. Lamarsh JR (1966) Introduction to Nuclear Reactor Theory. Addison-Wesley, Reading

2. Shibata K et al (2011) JENDL-4.0: a new library for nuclear science and engineering. J Nucl

Sci Tech 48:1

3. Chadwick MB et al (2011) ENDF/B-VII.1 nuclear data for science and technology: cross

sections, covariances, fission product yields and decay data. Nucl Data Sheets 112:2887

4. Koning A et al (ed) (2006) The JEFF-3.1 nuclear data library. JEFF report 21, NEA Data Bank

5. Trkov A, Herman M, Brown DA (eds) (2012) ENDF-6 Formats manual, data formats and

procedures for the evaluated nuclear data files ENDF/B-VI and ENDF/B-VII. CSEWG

Document ENDF-102, BNL-90365-2009 Rev. 2

6. Kobayashi K (1996) Nuclear reactor physics. Korona Press, Tokyo (in Japanese)

7. Okumura K, Kugo T, Kaneko K, Tsuchihashi K (2007) SRAC2006: a comprehensive neu-

tronics calculation code system, JAEA-Data/Code 2007-004. Japan Atomic Energy Agency,

Tokai-mura

8. Hazama T, Chiba G, Sato W, Numata K (2009) SLAROM-UF: ultra fine group cell calculation

code, JAEA-Review 2009-003. Japan Atomic Energy Agency, Tokai-mura

9. MacFarlane RE, Kahler AC (2010) Methods for processing ENDF/B-VII with NJOY. Nucl

Data Sheets 111:2739

10. Cullen DE (2012) PREPRO 2012 2012 ENDF/B pre-processing codes, IAEA-NDS-39. IAEA

Nuclear Data Section, Vienna


11. X-5 Monte Carlo Team (2003) MCNP: a general Monte Carlo N-particle transport code,

version 5, LA-UR-03-1987

12. Nagaya Y, Okumura K, Mori T, Nakagawa M (2005) MVP/GMVP II: general purpose Monte

Carlo codes for neutron and photon transport calculations based on continuous energy and

multigroup methods. JAERI 1348. Japan Atomic Energy Research Institute, Tokai-mura

13. Knott D, Yamamoto A (2010) Lattice physics computation. In: Handbook of nuclear engi-

neering, vol 2. Springer, New York, p 913

14. Nuclear Handbook Editorial Committee (2007) Nuclear handbook. Ohmsha Press, Tokyo

(in Japanese)

15. Okumura K (2007) COREBN: a core burn-up calculation module for SRAC2006, JAEA-Data/

Code 2007-003. Japan Atomic Energy Agency, Tokai-mura

16. Tasaka K (1997) DCHAIN: code for analysis of build-up and decay of nuclides, JAERI 1250.

Japan Atomic Energy Research Institute, Tokai-mura (in Japanese)

17. Croff AG (1983) ORIGEN2: a versatile computer code for calculating the nuclide composi-

tions and characteristics of nuclear materials. Nucl Tech 62:335

18. Lawrence RD (1986) Progress in nodal methods for the solution of the neutron diffusion and

transport equations. Prog Nucl Energy 17(3):271

19. Expert Research Committee on Improvement in Neutronics Calculation (2001) Status and

prospect of improvement in neutronics calculation. Atomic Energy Society of Japan, Tokyo

(in Japanese)

20. Greenman G, Smith KS, Henry AF (1979) Recent advances in an analytic nodal method for

static and transient reactor analysis. In: Proceedings of the computational methods in nuclear

engineering. ANS, Williamsburg, VA, pp 3–49, 23–25 Apr 1979

21. Okumura K (1998) MOSRA-light: high speed three-dimensional nodal diffusion code for

vector computers, JAERI-Data/Code 98-025. Japan Atomic Energy Research Institute, Tokai-

mura (in Japanese)

22. Ito N, Takeda T (1990) Three-dimensional multigroup diffusion code ANDEX based on nodal

method for cartesian geometry. J Nucl Sci Tech 27:350

23. Iwamoto T, Yamamoto M (1999) Advanced nodal methods of the few-group BWR core

simulator NEREUS. J Nucl Sci Tech 36:996

24. Argonne Code Center (1977) Benchmark problem book (11. Multi-dimensional (x-y-z) LWR

model), ANL-7416, Suppl. 2. Argonne National Laboratory, Illinois

25. Yamaji A (2005) Core and fuel design of the super LWR. Ph.D. Thesis, University of Tokyo

(in Japanese)

26. Oka Y et al (2009) Nuclear plant engineering, section 6–2: supercritical water-cooled reactor.

Ohmsha Press, Tokyo (in Japanese)

27. Oka Y, Koshizuka S, Ishiwatari Y, Yamaji A (2010) Super light water reactor and super fast

reactor. Springer, New York

28. Yi TT (2004) Startup and stability of a high temperature supercritical-pressure light water

reactor. Ph.D. Thesis, University of Tokyo

29. Ishiwatari Y (2006) Safety of the super LWR. Ph.D. Thesis, University of Tokyo (in Japanese)

30. Suzuki M, Saitou H (2006) Light water reactor fuel analysis code FEMAXI-6(Ver.1) �Detailed Structure and User’s Manual JAEA-Data/Code 2005-003). Japan Atomic Energy

Agency, Tokai-mura

31. Oka Y, Suzuki K (2008) Nuclear reactor dynamics and plant control. Ohmsha Press, Tokyo

(in Japanese)

32. The Japan Society of Mechanical Engineers (1986) Heat transfer, Rev. 4. The Japan Society of

Mechanical Engineers (in Japanese)


http://www.springer.com/978-4-431-54897-3

Chapter 2 Nuclear Reactor Calculations

Documents

Transcript of Chapter 2 Nuclear Reactor Calculations